PSAT Math Flashcards: Lines Angles And Triangles

What this deck covers

This deck focuses on Lines Angles And Triangles, giving you a quick way to review the definitions, rules, and examples that matter most for PSAT Math.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

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

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

Flashcard 2: What is the sum of the interior angles of a triangle?

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

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

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

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

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

Flashcard 5: What is the definition of supplementary angles?

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

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

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

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

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

Flashcard 8: What is the sum of the interior angles of a triangle?

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Flashcard 31: What is the definition of complementary angles in degrees?

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Flashcard 72: What is the sum of the angles around a point?

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

Flashcard 73: What is the sum of the interior angles of a triangle?

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Flashcard 86: What is the exterior angle theorem for a triangle?

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

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

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

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

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

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

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

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

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

Flashcard 91: What is the definition of complementary angles in degrees?

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

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

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

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

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

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

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

Flashcard 95: What is the converse of the Pythagorean Theorem?

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

Flashcard 96: What is the definition of supplementary angles in degrees?

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

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

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

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

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

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

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

Flashcard 100: Identify the missing angle: vertical to a $37^\circ$ angle is <span class="fill-in-blank">&nbsp;</span>^\circ.

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