Lines, Angles, & Triangles - PSAT Math

1. What does the Triangle Inequality Theorem state for side lengths $a$, $b$, and $c$?

   $a+b>c$, $a+c>b$, and $b+c>a$. The sum of any two sides must exceed the third side's length.

2. What is the sum of the interior angles of a triangle?

   $180^\circ$. Fundamental property: all triangles have angle sum of $180^\circ$.

3. Find $x$ if a triangle has angles $2x$, $3x$, and $70^\circ$.

   $x=22$. Triangle angles sum: $2x+3x+70=180$; solve $5x=110$.

4. Identify $x$ if a triangle has angles $50^\circ$, $60^\circ$, and $x^\circ$.

   $x=70^\circ$. Use $50^\circ + 60^\circ + x = 180^\circ$.

5. What is the definition of supplementary angles?

   Two angles with sum $180^\circ$. Supplementary means they complete a straight angle together.

6. What is the relationship between the exterior angle of a triangle and the two remote interior angles?

   Exterior angle equals their sum. The exterior angle theorem for triangles.

7. Find $x$ if corresponding angles are congruent and measure $(7x+5)^\circ$ and $(3x+45)^\circ$.

   $x=10$. Set angles equal: $7x+5=3x+45$, so $4x=40$.

8. What is the sum of the interior angles of a triangle?

   $180^\circ$. This fundamental property holds for all triangles.

9. What is $x$ if a triangle has angles $35^\circ$, $60^\circ$, and $x^\circ$?

   $x=85^\circ$. Triangle angles sum to $180°$: $35+60+x=180$.

10. What condition guarantees two triangles are similar using two pairs of equal angles?

    $\text{AA similarity}$. Two pairs of equal angles guarantee triangles are similar.

11. What is the sum of the exterior angles of any polygon (one at each vertex)?

    $360^\circ$. Each exterior angle pairs with an interior angle to make $180^\circ$.

12. Identify whether side lengths $6$, $8$, and $10$ form a right triangle.

    Yes, because $6^2+8^2=10^2$. Check if $36 + 64 = 100$ (Pythagorean theorem).

13. What is the relationship between same-side interior angles when lines are parallel?

    They are supplementary (sum to $180^\circ$). Same-side interior angles between parallel lines always sum to $180^\circ$.

14. Identify whether side lengths $4$, $9$, and $13$ can form a triangle.

    No, because $4+9\not>13$. Triangle inequality fails: $4+9=13$, not greater than $13$.

15. What is the angle relationship called when two angles sum to $180^\circ$?

    Supplementary angles. Two angles that add up to $180^\circ$ form a straight line together.

16. What is the sum of the angles that form a linear pair?

    $180^\circ$. Linear pairs are supplementary by definition.

17. What is the measure of each interior angle of a regular $n$-gon?

    $\frac{(n-2)\cdot 180^\circ}{n}$. Divide total interior angle sum by $n$ for regular polygons.

18. Find the hypotenuse if a $45^\circ$-$45^\circ$-$90^\circ$ triangle has leg length $7$.

    $7\sqrt{2}$. In $45^\circ$-$45^\circ$-$90^\circ$, hypotenuse is leg times $\sqrt{2}$.

19. What is the sum of the measures of a linear pair of adjacent angles?

    $180^\circ$. Adjacent angles on a straight line sum to a straight angle.

20. What is the measure of each angle in a linear pair if one angle is $x^\circ$?

    The other angle is $(180-x)^\circ$. Linear pairs are supplementary, so they sum to $180^\circ$.

21. What are the side ratios for a $45^\circ$-$45^\circ$-$90^\circ$ triangle?

    $1:1:\sqrt{2}$. Isosceles right triangle has legs equal and hypotenuse $\sqrt{2}$ times a leg.

22. Find $x$ if vertical angles have measures $(5x-15)^\circ$ and $(3x+25)^\circ$.

    $x=20$. Set $5x-15 = 3x+25$ since vertical angles are equal.

23. What is the definition of an exterior angle of a triangle?

    An angle formed by extending one side of a triangle. Forms a linear pair with the adjacent interior angle.

24. What is the measure of the exterior angle at a triangle vertex if the two remote interior angles are $50^\circ$ and $65^\circ$?

    $115^\circ$. Exterior angle equals sum of non-adjacent interior angles.

25. What is the condition for two non-vertical lines with slopes $m_1$ and $m_2$ to be parallel?

    $m_1=m_2$. Parallel lines have identical slopes and never intersect.

26. Find $x$ if two angles are complementary and one angle is $35^\circ$ while the other is $x^\circ$.

    $x=55$. Complementary angles sum to $90^\circ$, so $x = 90 - 35$.

27. What is the relationship between adjacent angles that form a linear pair?

    They are supplementary. Adjacent angles on a straight line sum to $180^\circ$.

28. What does the Pythagorean Theorem state for a right triangle with legs $a$, $b$ and hypotenuse $c$?

    $a^2+b^2=c^2$. In right triangles, legs squared sum to hypotenuse squared.

29. Find $x$ if a triangle has angles $50^\circ$, $60^\circ$, and $x^\circ$.

    $x=70^\circ$. Triangle angles sum to $180°$: $50+60+x=180$.

30. What is the sum of the interior angle measures of a triangle?

    $180^\circ$. The three interior angles of any triangle sum to a straight angle.

31. What is the definition of complementary angles in degrees?

    $m\angle A + m\angle B = 90^\circ$. Two angles that sum to $90^\circ$ are complementary.

32. What is the sum of the interior angles of a convex $n$-gon?

    $(n-2)\cdot 180^\circ$. Each triangle adds $180^\circ$; an $n$-gon forms $n-2$ triangles.

33. What is $x$ if $x$ is vertical to a $68^\circ$ angle formed by intersecting lines?

    $68^\circ$. Vertical angles are always congruent.

34. What is the sum of the measures of angles around a point?

    $360^\circ$. A complete rotation around any point equals one full circle.

35. Find the missing angle if two angles in a triangle are $35^\circ$ and $65^\circ$.

    $80^\circ$. Triangle angles sum to $180^\circ$: $35+65+x=180$.

36. What is the angle relationship between two lines with slopes $m_1$ and $m_2$ if the lines are perpendicular?

    $m_1m_2=-1$. Product of perpendicular slopes always equals $-1$.

37. What is the relationship between adjacent angles that form a linear pair?

    They are supplementary (sum to $180^\circ$). Adjacent angles on a straight line must total a straight angle.

38. Identify the value of $x$ if two angles form a linear pair: $(3x+10)^\circ$ and $(2x+20)^\circ$.

    $x=30$. Linear pair angles sum to $180^\circ$: $(3x+10)+(2x+20)=180$.

39. Find $x$ if vertical angles are labeled $(3x+10)^\circ$ and $(5x-14)^\circ$.

    $x=12$. Vertical angles are equal: $3x + 10 = 5x - 14$, solving gives $x = 12$.

40. Find the third angle of a triangle if two angles are $35^\circ$ and $65^\circ$.

    $80^\circ$. Triangle angles sum to $180^\circ$: $35+65+x=180$.

41. Identify the angle measure: If two angles are supplementary and one is $112^\circ$, what is the other?

    $68^\circ$. Since $112^\circ + 68^\circ = 180^\circ$.

42. What is the measure relationship of vertical angles formed by intersecting lines?

    They are congruent: measures are equal. Opposite angles formed by two intersecting lines are always equal.

43. Identify the angle: If two lines intersect and one angle is $47^\circ$, what is its vertical angle?

    $47^\circ$. Vertical angles are congruent, so both equal $47^\circ$.

44. Find $x$ if an isosceles triangle has base angles $x^\circ$ and vertex angle $40^\circ$.

    $x=70^\circ$. Base angles equal, so $2x + 40^\circ = 180^\circ$.

45. Identify the angle measure: If two angles are complementary and one is $37^\circ$, what is the other?

    $53^\circ$. Since $37^\circ + 53^\circ = 90^\circ$.

46. What is the equation of a line with slope $m$ passing through $(x_1,y_1)$?

    $y-y_1=m(x-x_1)$. Point-slope form: substitute known point and slope.

47. What is the condition for two non-vertical lines to be perpendicular using slopes $m_1$ and $m_2$?

    $m_1m_2=-1$. Perpendicular slopes multiply to negative one.

48. What is the definition of an equilateral triangle in terms of angles?

    All three angles are $60^\circ$. All sides equal means all angles equal; $180^\circ \div 3 = 60^\circ$.

49. Identify the value of $x$ if a linear pair measures $(2x+15)^\circ$ and $(5x-6)^\circ$.

    $x=\frac{171}{7}$. Linear pair sums to $180°$: $(2x+15)+(5x-6)=180$.

50. Identify whether side lengths $3$, $4$, and $8$ can form a triangle.

    No, because $3+4\le 8$. Violates triangle inequality since $3+4=7<8$.

51. What is the third angle measure if a triangle has angles $50^\circ$ and $60^\circ$?

    $70^\circ$. Since $50^\circ + 60^\circ + 70^\circ = 180^\circ$.

52. Find $x$ if same-side interior angles are $(3x+20)^\circ$ and $(2x+10)^\circ$ for parallel lines.

    $x=30$. Same-side interior angles sum to $180°$: $(3x+20)+(2x+10)=180$.

53. What is the special right triangle ratio for a $30^\circ$-$60^\circ$-$90^\circ$ triangle?

    $1:\sqrt{3}:2$. Sides opposite $30^\circ$, $60^\circ$, $90^\circ$ are in ratio $1:\sqrt{3}:2$.

54. What is the hypotenuse length if a right triangle has legs $6$ and $8$?

    $10$. By Pythagorean theorem: $\sqrt{6^2 + 8^2} = \sqrt{100} = 10$.

55. Find $x$: In an isosceles triangle, the vertex angle is $40^\circ$ and each base angle is $x^\circ$. What is $x$?

    $70^\circ$. Base angles equal: $40+2x=180$, so $2x=140$, thus $x=70$.

56. What is the relationship between same-side (consecutive) interior angles when lines are parallel?

    Same-side interior angles are supplementary. Interior angles on same side of transversal sum to $180°$ for parallel lines.

57. Identify whether $7$, $10$, and $12$ can form a triangle using the Triangle Inequality.

    Yes, they can form a triangle. Check: $7+10>12$, $7+12>10$, and $10+12>7$ all true.

58. Find the missing side $c$ if a right triangle has legs $6$ and $8$.

    $c=10$. Use Pythagorean theorem: $c^2=6^2+8^2=36+64=100$.

59. What is the condition for two non-vertical lines with slopes $m_1$ and $m_2$ to be perpendicular?

    $m_1m_2=-1$. Perpendicular lines have slopes that multiply to negative one.

60. What is the angle sum around a point in degrees?

    $360^\circ$. All angles meeting at a single point form a complete rotation.

61. What is the Triangle Inequality Theorem stated using side lengths $a$, $b$, and $c$?

    $a+b>c$, $a+c>b$, and $b+c>a$. The sum of any two sides must exceed the third side.

62. What is the relationship between slopes of parallel non-vertical lines?

    $m_1=m_2$. Parallel lines have identical slopes.

63. What is the definition of an isosceles triangle in terms of side lengths?

    A triangle with at least two congruent sides. The equal sides can be any two of the three sides.

64. Find $x$ if two angles form a linear pair and one is $112^\circ$ while the other is $x^\circ$.

    $x=68$. Linear pairs sum to $180^\circ$, so $x = 180 - 112$.

65. What is the triangle inequality for side lengths $a$, $b$, and $c$?

    $a+b>c$, $a+c>b$, and $b+c>a$. Sum of any two sides must exceed the third side.

66. What are the side lengths in a $45^\circ$-$45^\circ$-$90^\circ$ triangle if each leg is $x$?

    $x,\ x,\ x\sqrt{2}$. Isosceles right triangle has ratio $1:1:\sqrt{2}$.

67. Find $x$ if two complementary angles have measures $x^\circ$ and $(3x-20)^\circ$.

    $x=27.5$. Set $x + (3x-20) = 90$ and solve for $x$.

68. What is the special right triangle ratio for a $45^\circ$-$45^\circ$-$90^\circ$ triangle?

    $1:1:\sqrt{2}$. Isosceles right triangle has sides in ratio $1:1:\sqrt{2}$.

69. Find $x$ if angles in a linear pair are $(3x+10)^\circ$ and $(2x+20)^\circ$.

    $x=30$. Linear pair: $(3x+10)+(2x+20)=180$, so $5x+30=180$.

70. What is the relationship of two angles whose measures add to $180^\circ$?

    They are supplementary angles. Two angles that sum to a straight angle form a supplementary pair.

71. Find $x$: If two angles form a linear pair and are $(3x+10)^\circ$ and $(2x+20)^\circ$, what is $x$?

    $30$. Linear pair: $(3x+10)+(2x+20)=180$, so $5x+30=180$, thus $x=30$.

72. What is the sum of the angles around a point?

    $360^\circ$. A complete rotation around any point equals $360^\circ$.

73. What is the sum of the interior angles of a triangle?

    $180^\circ$. The three interior angles of any triangle always sum to this value.

74. What is the slope of a line perpendicular to a line with slope $\frac{2}{3}$?

    $-\frac{3}{2}$. Perpendicular slope is negative reciprocal: $-1÷\frac{2}{3}=-\frac{3}{2}$.

75. What is the Triangle Inequality Theorem stated using side lengths $a$, $b$, and $c$?

    $a+b>c$, $a+c>b$, and $b+c>a$. Each side must be less than the sum of the other two.

76. What is the relationship between vertical angles formed by two intersecting lines?

    Vertical angles are congruent. Opposite angles formed by intersecting lines are equal.

77. What is the angle sum of a straight line (a linear pair) in degrees?

    $180^\circ$. Adjacent angles forming a straight line always sum to this value.

78. Find the missing angle if a triangle has angles $35^\circ$ and $65^\circ$.

    $80^\circ$. Use $35 + 65 + x = 180$ to find missing angle.

79. Find $x$ if same-side interior angles are $(2x+20)^\circ$ and $(3x+10)^\circ$ on parallel lines.

    $x=30$. Set $(2x+20) + (3x+10) = 180$ and solve.

80. Find $x$ if two angles form a linear pair and have measures $(3x+10)^\circ$ and $(2x+20)^\circ$.

    $x=30$. Linear pair: $(3x+10)+(2x+20)=180$, so $5x+30=180$.

81. What is the measure of a straight angle in degrees?

    $180^\circ$. A straight line forms a flat angle of half a full rotation.

82. What is the relationship between same-side interior angles when lines are parallel?

    They are supplementary: sum is $180^\circ$. Also called co-interior or consecutive interior angles.

83. Identify the hypotenuse length $c$ of a right triangle with legs $6$ and $8$.

    $c=10$. Apply Pythagorean theorem: $c=\sqrt{6^2+8^2}=\sqrt{36+64}=10$.

84. What is the Pythagorean Theorem for a right triangle with legs $a$, $b$ and hypotenuse $c$?

    $a^2+b^2=c^2$. Relates legs and hypotenuse in right triangles.

85. What is the measure relationship of angles that form a linear pair?

    They are supplementary: sum is $180^\circ$. Adjacent angles on a straight line always total $180^\circ$.

86. What is the exterior angle theorem for a triangle?

    Exterior angle equals sum of two remote interior angles. Exterior angle extends one side beyond the triangle.

87. What is the slope formula for the line through points $(x_1,y_1)$ and $(x_2,y_2)$?

    $m=\frac{y_2-y_1}{x_2-x_1}$. Rise over run: change in y divided by change in x.

88. What equation relates two adjacent angles that form a linear pair?

    $m\angle 1 + m\angle 2 = 180^\circ$. Adjacent angles on a straight line always sum to a straight angle.

89. What is the slope of a vertical line (in terms of $m$)?

    $m$ is undefined. Vertical lines have no run, so division by zero is undefined.

90. What is the sum of the measures of the exterior angles of any convex polygon (one at each vertex)?

    $360^\circ$. Each exterior angle pairs with interior to make $180°$; sum equals full rotation.

91. What is the definition of complementary angles in degrees?

    Two angles whose measures sum to $90^\circ$. Complementary means "completing to a right angle."

92. Find the hypotenuse $c$ of a right triangle with legs $6$ and $8$.

    $c=10$. Using Pythagorean theorem: $c = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100}$.

93. What is the relationship between slopes of perpendicular non-vertical lines?

    $m_1m_2=-1$. Perpendicular lines have slopes whose product is negative one.

94. What is the relationship between same-side (consecutive) interior angles for parallel lines cut by a transversal?

    They are supplementary; sum is $180^\circ$. Interior angles on the same side of the transversal add to a straight angle.

95. What is the converse of the Pythagorean Theorem?

    If $a^2+b^2=c^2$, then the triangle is right. Tests if a triangle is right-angled using side lengths.

96. What is the definition of supplementary angles in degrees?

    $m\angle A + m\angle B = 180^\circ$. Two angles that sum to $180^\circ$ are supplementary.

97. What is the Triangle Inequality in terms of side lengths $a$, $b$, and $c$?

    $a+b>c$, $a+c>b$, and $b+c>a$. Sum of any two sides must exceed the third side.

98. Identify the value of $x$ if two vertical angles measure $(3x+10)^\circ$ and $(5x-30)^\circ$.

    $x=20$. Set vertical angles equal: $3x+10=5x-30$; solve for $x$.

99. Find the hypotenuse $c$ if a right triangle has legs $6$ and $8$.

    $c=10$. Pythagorean theorem: $c^2=6^2+8^2=36+64=100$.

100. Identify the missing angle: vertical to a $37^\circ$ angle is $\_\_\_^\circ$.

     $37^\circ$. Vertical angles are congruent, so they have the same measure.

101. What is the relationship between a linear pair of adjacent angles?

     They are supplementary (sum to $180^\circ$). Adjacent angles on a straight line sum to $180°$.

102. What is the Triangle Inequality Theorem for side lengths $a$, $b$, and $c$?

     $a+b>c$, $a+c>b$, and $b+c>a$. Sum of any two sides must exceed the third side for a valid triangle.

103. What is the special right triangle ratio for a $30^\circ$-$60^\circ$-$90^\circ$ triangle?

     $x,x\sqrt{3},2x$. Sides opposite $30°$, $60°$, $90°$ angles respectively.

104. Find the hypotenuse $c$ of a right triangle with legs $6$ and $8$.

     $c=10$. Apply Pythagorean theorem: $c=\sqrt{6^2+8^2}=\sqrt{100}$.

105. What are the side lengths in a $30^\circ$-$60^\circ$-$90^\circ$ triangle if the short leg is $x$?

     $x,\ x\sqrt{3},\ 2x$. Special triangle with side ratio $1:\sqrt{3}:2$.

106. What relationship do vertical angles formed by two intersecting lines have?

     They are congruent: $m\angle 1 = m\angle 2$. Opposite angles formed by intersecting lines always have equal measures.

107. Can lengths $3$, $4$, and $8$ form a triangle? Answer $\text{Yes}$ or $\text{No}$.

     $\text{No}$. $3 + 4 = 7 < 8$, violating the triangle inequality.

108. Find the missing side $c$ if a right triangle has legs $6$ and $8$.

     $c=10$. Pythagorean theorem: $c=\sqrt{6^2+8^2}=\sqrt{100}=10$.

109. What is the sum of the interior angles of a triangle?

     $180^\circ$. Fundamental property: all triangles have angle sum of $180°$.

110. What is the relationship between alternate interior angles when two parallel lines are cut by a transversal?

     Alternate interior angles are congruent. Interior angles on opposite sides of transversal are equal.

111. What is the exterior angle theorem for a triangle with remote interior angles $A$ and $B$?

     $m\angle \text{ext}=m\angle A+m\angle B$. An exterior angle equals the sum of the two non-adjacent interior angles.

112. What is the Triangle Inequality Theorem for side lengths $a$, $b$, and $c$?

     $a+b>c$, $a+c>b$, and $b+c>a$. Sum of any two sides must exceed the third side.

113. What is the Triangle Inequality Theorem stated using side lengths $a$, $b$, and $c$?

     $a+b>c,\ a+c>b,\ b+c>a$. Sum of any two sides must exceed the third side.

114. What is the angle sum of any triangle in degrees?

     $180^\circ$. All interior angles of any triangle always sum to this value.

115. What is the value of $x$ if an exterior angle is $125^\circ$ and one remote interior angle is $55^\circ$?

     $70^\circ$. Exterior angle equals sum of remote interiors: $55^\circ + 70^\circ = 125^\circ$.

116. What is the relationship between adjacent angles on a straight line?

     They are supplementary; sum is $180^\circ$. Adjacent angles on a line form a straight angle.

117. Identify the missing angle: If two angles are supplementary and one is $68^\circ$, what is the other?

     $112^\circ$. Supplementary angles sum to $180^\circ$, so $180^\circ - 68^\circ = 112^\circ$.

118. What is the sum of the measures of the exterior angles of a triangle (one at each vertex)?

     $360^\circ$. One exterior angle at each vertex always sums to a full rotation.

119. Find $x$ if same-side interior angles are $(7x+5)^\circ$ and $(3x+15)^\circ$ for parallel lines.

     $x=16$. Same-side interior: $(7x+5)+(3x+15)=180$, so $10x+20=180$.

120. What is true about the base angles of an isosceles triangle?

     The base angles are congruent. Angles opposite the congruent sides are equal.

121. What is the relationship between supplementary angles?

     Their measures sum to $180^\circ$. Supplementary angles form a straight angle together.

122. Find $x$ if vertical angles have measures $(5x-7)^\circ$ and $(3x+9)^\circ$.

     $x=8$. Vertical angles equal: $5x-7=3x+9$, so $2x=16$.

123. In a triangle, what is the relationship between side length and opposite angle size?

     Longer side is opposite larger angle. In any triangle, larger angles are opposite longer sides.

124. If corresponding sides of similar triangles have ratio $\frac{3}{5}$, what is the ratio of their areas?

     $\frac{9}{25}$. Area ratio equals the square of the side ratio: $(\frac{3}{5})^2$.

125. Find $x$ if two angles form a linear pair and have measures $3x+10$ and $5x-6$ degrees.

     $x=22$. Linear pair: $(3x+10)+(5x-6)=180$; solve $8x+4=180$.

126. What is the relationship between same-side interior angles when lines are parallel?

     Same-side interior angles are supplementary. Interior angles on same side of transversal sum to $180^\circ$ when parallel.

127. What is the measure of each angle in an equilateral triangle?

     Each angle is $60^\circ$. All angles equal in an equilateral triangle: $180^\circ \div 3$.

128. What is the area formula for a triangle with base $b$ and height $h$?

     $A=\frac{1}{2}bh$. Half the product of base and perpendicular height.

129. What is $x$ if $x$ and $45^\circ$ form a linear pair?

     $135^\circ$. Linear pairs are supplementary: $45^\circ + 135^\circ = 180^\circ$.

130. What is the measure of a full rotation (one complete turn) in degrees?

     $360^\circ$. A complete circle or full turn measures this many degrees.

131. What is the relationship between base angles in an isosceles triangle?

     The base angles are congruent. In a triangle with two equal sides, the opposite angles are equal.

132. What is the relationship between the base angles of an isosceles triangle?

     The base angles are congruent. Angles opposite equal sides are equal.

133. Find $x$ if two vertical angles have measures $(3x+10)^\circ$ and $(5x-30)^\circ$.

     $x=20$. Set vertical angles equal: $3x+10=5x-30$, then solve for $x$.

134. What is the definition of supplementary angles in degrees?

     Two angles whose measures sum to $180^\circ$. Supplementary means "completing to a straight angle."

135. What is the angle sum of a triangle in degrees?

     $180^\circ$. Sum of interior angles in any triangle is always $180^\circ$.

136. What is the condition for two non-vertical lines to be parallel using slopes $m_1$ and $m_2$?

     $m_1=m_2$. Parallel lines have equal slopes.

137. What is the angle measure of a straight angle?

     $180^\circ$. A straight line forms a flat angle of exactly half a full rotation.

138. What is the sum of the interior angles of any triangle?

     $180^\circ$. The three interior angles of any triangle always add to this value.

139. What is the relationship between adjacent angles that form a linear pair?

     They are supplementary. Adjacent angles on a line always sum to $180^\circ$.

140. What is the sum of the interior angles of a triangle?

     $180^\circ$. Any triangle's three angles always total a straight angle.

141. What is the measure of each interior angle of an equilateral triangle?

     $60^\circ$. All three equal angles divide $180°$ equally: $180° ÷ 3 = 60°$.

142. What are the side ratios for a $30^\circ$-$60^\circ$-$90^\circ$ triangle?

     $1:\sqrt{3}:2$. Short leg : long leg : hypotenuse in a 30-60-90 triangle.

143. Find $x$ if two angles form a linear pair and their measures are $(3x+10)^\circ$ and $(2x+20)^\circ$.

     $x=30$. Linear pair: $(3x+10)+(2x+20)=180$, so $5x+30=180$.

144. What is the relationship between complementary angles?

     Their measures sum to $90^\circ$. Complementary angles form a right angle together.

145. What is the relationship between an exterior angle of a triangle and the two remote interior angles?

     $m\angle_{\text{ext}} = m\angle_1 + m\angle_2$. An exterior angle equals the sum of the two non-adjacent interior angles.

146. What is the Triangle Inequality Theorem for side lengths $a$, $b$, and $c$?

     $a+b>c$, $a+c>b$, and $b+c>a$. The sum of any two sides must exceed the third side.

147. What is the angle relationship called when two angles sum to $90^\circ$?

     Complementary angles. Two angles that add up to $90^\circ$ form a right angle together.

148. What is the relationship between corresponding angles when lines are parallel?

     They are congruent (equal in measure). Parallel lines cut by a transversal create equal corresponding angles.

149. What is the exterior angle measure if the interior angle is $125^\circ$?

     $55^\circ$. Exterior and interior angles are supplementary: $180^\circ - 125^\circ$.

150. What is the sum of the measures of angles around a point?

     $360^\circ$. A complete rotation around any point equals one full turn.

151. Find $x$ if a linear pair has measures $(2x+15)^\circ$ and $(x+30)^\circ$.

     $x=45$. Linear pair sums to $180°$: $(2x+15)+(x+30)=180$.

152. Find $x$ if a triangle has angles $50^\circ$, $60^\circ$, and $x^\circ$.

     $x=70$. Triangle angles sum to $180^\circ$, so $x = 180 - 50 - 60$.

153. Which angle pairs are congruent when a transversal cuts parallel lines: corresponding or same-side interior?

     Corresponding angles are congruent. Angles in matching positions at parallel lines are equal.

154. What is the relationship between alternate interior angles when lines are parallel?

     Alternate interior angles are congruent. Interior angles on opposite sides of transversal are equal when parallel.

155. Find the hypotenuse: In a right triangle with legs $6$ and $8$, what is the hypotenuse length?

     $10$. By Pythagorean theorem: $c^2=6^2+8^2=36+64=100$, so $c=10$.

156. What is the relationship between same-side interior angles when lines are parallel?

     Same-side interior angles are supplementary. Interior angles on same side sum to $180^\circ$.

157. What is the Exterior Angle Theorem for a triangle with remote interior angles $A$ and $B$?

     Exterior angle $=A+B$. The exterior angle equals the sum of non-adjacent interior angles.

158. Find the missing side $c$ in a right triangle with legs $6$ and $8$.

     $c=10$. Pythagorean theorem: $c^2=6^2+8^2=36+64=100$.

159. What is the definition of complementary angles in degrees?

     Two angles whose measures sum to $90^\circ$. Together they form a right angle.

160. What theorem states that an exterior angle of a triangle equals the sum of two remote interior angles?

     Exterior Angle Theorem. The exterior angle equals the sum of the two non-adjacent interior angles.

161. If the short leg of a $30^\circ$-$60^\circ$-$90^\circ$ triangle is $7$, what is the hypotenuse?

     $14$. In 30-60-90 triangles, hypotenuse is twice the short leg.

162. Find $x$ if corresponding angles are labeled $(3x+10)^\circ$ and $85^\circ$ on parallel lines.

     $x=25$. Set $3x + 10 = 85$ and solve for $x$.

163. What inequality must side lengths $a$, $b$, and $c$ satisfy to form a triangle?

     $a + b > c$, $a + c > b$, $b + c > a$. Triangle Inequality: sum of any two sides exceeds the third.

164. What is the relationship between vertical angles formed by intersecting lines?

     Vertical angles are congruent. Opposite angles formed by two intersecting lines are always equal.

165. Identify whether side lengths $3$, $4$, and $8$ can form a triangle.

     No, because $3+4\le 8$. Violates triangle inequality since $3+4=7<8$.

166. What is the relationship of two angles whose measures add to $90^\circ$?

     They are complementary angles. Two angles that sum to a right angle form a complementary pair.

167. Find the hypotenuse $c$ of a right triangle with legs $6$ and $8$ using $a^2+b^2=c^2$.

     $c=10$. $6^2+8^2=36+64=100$, so $c=\sqrt{100}=10$.

168. What is the sum of the interior angles of a triangle?

     $180^\circ$. All triangles have angle sum of $180^\circ$.

169. What is the slope of a horizontal line (in terms of $m$)?

     $m=0$. Horizontal lines have no rise, so slope is zero.

170. Find $x$ if alternate interior angles are $(6x+6)^\circ$ and $90^\circ$ for parallel lines.

     $x=14$. Alternate interior angles equal: $6x+6=90$, so $6x=84$.

171. Find $x$: In a triangle, angles are $50^\circ$, $60^\circ$, and $x^\circ$. What is $x$?

     $70^\circ$. Triangle angles sum to $180^\circ$: $50+60+x=180$, so $x=70$.

172. Find $x$ if vertical angles are labeled $(4x-5)^\circ$ and $(2x+25)^\circ$.

     $x=15$. Vertical angles are equal: $4x-5=2x+25$, so $2x=30$.

173. What is the relationship between alternate interior angles when lines are parallel?

     They are congruent. Alternate interior angles are equal when lines are parallel.

174. Find $x$ if a triangle has angles $50^\circ$, $60^\circ$, and $x^\circ$.

     $x=70^\circ$. Triangle angles sum to $180°$: $50+60+x=180$.

175. Find the missing leg $b$ of a right triangle with hypotenuse $13$ and leg $5$.

     $b=12$. Use $a^2+b^2=c^2$: $5^2+b^2=13^2$, so $b^2=144$.

176. Find $x$ if vertical angles have measures $7x-4$ and $3x+20$ degrees.

     $x=6$. Vertical angles equal: $7x-4=3x+20$; solve $4x=24$.

177. What is the relationship between same-side interior angles when two parallel lines are cut by a transversal?

     They are supplementary; sum is $180^\circ$. Interior angles on same side of transversal add to $180^\circ$.

178. Find $x$ if vertical angles measure $(4x-5)^\circ$ and $(2x+15)^\circ$.

     $x=10$. Vertical angles equal: $4x-5=2x+15$, so $2x=20$.

179. What is the special right triangle ratio for a $45^\circ$-$45^\circ$-$90^\circ$ triangle?

     $x,x,x\sqrt{2}$. Isosceles right triangle: legs equal, hypotenuse is leg times $\sqrt{2}$.

180. What is the measure relationship for angles that form a linear pair?

     $m\angle 1 + m\angle 2 = 180^\circ$. Adjacent angles on a straight line sum to $180^\circ$.

181. What is the definition of complementary angles?

     Two angles with sum $90^\circ$. Complementary means they complete a right angle together.

182. Find the third angle of a triangle with angles $50^\circ$ and $60^\circ$.

     $70^\circ$. Use $180^\circ - 50^\circ - 60^\circ = 70^\circ$.

183. What is the sum of the interior angles of a triangle in degrees?

     $180^\circ$. The angle sum property of triangles is always $180^\circ$.

184. Find $c$ if a right triangle has legs $5$ and $12$ and hypotenuse $c$.

     $13$. Apply Pythagorean theorem: $c = \sqrt{5^2 + 12^2} = \sqrt{169} = 13$.

185. Find $x$ if corresponding angles are $(5x-8)^\circ$ and $67^\circ$ for parallel lines.

     $x=15$. Corresponding angles equal: $5x-8=67$, so $5x=75$.

186. Find $x$ if two supplementary angles have measures $(2x+10)^\circ$ and $(x+20)^\circ$.

     $x=50$. Set $(2x+10) + (x+20) = 180$ and solve for $x$.

187. What is the sum of the measures of the interior angles of a triangle?

     $180^\circ$. Triangle angle sum theorem.

188. What is the definition of supplementary angles in degrees?

     Two angles whose measures sum to $180^\circ$. Together they form a straight angle.

189. What is the degree measure of angles around a point (a full turn)?

     $360^\circ$. A complete rotation around a point equals one full circle.

190. What is the Pythagorean Theorem for a right triangle with legs $a,b$ and hypotenuse $c$?

     $a^2+b^2=c^2$. Relates squares of sides in a right triangle.

191. What is the relationship between corresponding angles when two parallel lines are cut by a transversal?

     Corresponding angles are congruent. Same position angles at each intersection are equal.

192. Identify the value of $x$ if corresponding angles are $(4x+5)^\circ$ and $97^\circ$ with parallel lines.

     $x=23$. Set corresponding angles equal: $4x+5=97$; solve for $x$.

193. Find the exterior angle if the two remote interior angles are $35^\circ$ and $65^\circ$.

     $100^\circ$. Apply the Exterior Angle Theorem: $35^\circ + 65^\circ = 100^\circ$.

194. Find the exterior angle if the two remote interior angles are $40^\circ$ and $75^\circ$.

     $115^\circ$. Exterior angle equals sum of non-adjacent interior angles.

195. What is the relationship between same-side (consecutive) interior angles when lines are parallel?

     They are supplementary: sum is $180^\circ$. Interior angles on the same side of a transversal add to $180°$ for parallel lines.

196. What is the relationship between vertical angles formed by intersecting lines?

     Vertical angles are congruent. Opposite angles formed by intersecting lines have equal measures.

197. Find $x$ if vertical angles have measures $(5x-15)^\circ$ and $(2x+45)^\circ$.

     $x=20$. Vertical angles equal: $5x-15=2x+45$, so $3x=60$.

198. What is the degree measure of a straight angle?

     $180^\circ$. A straight line forms a flat angle of half a rotation.

199. Find $x$ if a triangle has angles $50^\circ$, $(x+10)^\circ$, and $(2x)^\circ$.

     $x=40$. Triangle angles sum to $180^\circ$: $50+(x+10)+2x=180$.

200. What is the measure relationship for vertical angles?

     $m\angle 1 = m\angle 2$. Opposite angles formed by intersecting lines are equal.

201. Find $x$ if corresponding angles are $(4x-5)^\circ$ and $87^\circ$ for parallel lines cut by a transversal.

     $x=23$. Corresponding angles are equal: $4x-5=87$, so $4x=92$.

202. Find the exterior angle measure if the two remote interior angles are $35^\circ$ and $65^\circ$.

     $100^\circ$. Exterior angle equals sum: $35^\circ+65^\circ=100^\circ$.

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