Lines, Angles, and Triangles - Quantitative Reasoning

1. What are the side-length ratios of a $30^\circ$-$60^\circ$-$90^\circ$ triangle?

   $1:\sqrt{3}:2$. In a $30^\circ$-$60^\circ$-$90^\circ$ triangle, sides are in the ratio shortest to middle to hypotenuse as $1 : \sqrt{3} : 2$.

2. What is the sum of the interior angles of an $n$-gon in degrees?

   $(n-2)\cdot 180^\circ$. The formula derives from dividing the polygon into $n-2$ triangles, each contributing $180^\circ$.

3. What is the relationship between complementary angles?

   They sum to $90^\circ$. Complementary angles are defined as two angles whose measures add up to $90^\circ$.

4. What is the base-angle theorem for an isosceles triangle?

   Angles opposite equal sides are equal. In an isosceles triangle, the angles opposite the equal sides are congruent.

5. What is the measure of an exterior angle of a triangle in terms of its two remote interior angles?

   Exterior angle equals sum of the two remote interior angles. The exterior angle theorem states that an exterior angle equals the sum of the two non-adjacent interior angles.

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

   $a^2+b^2=c^2$. The theorem relates the sides of a right triangle, where the square of the hypotenuse equals the sum of the squares of the legs.

7. 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 theorem states that the sum of any two sides must exceed the third to form a triangle.

8. What is the relationship between vertical angles?

   They are congruent (equal in measure). Vertical angles are formed by intersecting lines and are equal because they are opposite each other.

9. What is the relationship between side lengths and opposite angles in any triangle?

   Longer side is opposite larger angle. In any triangle, the largest angle is opposite the longest side, and vice versa, by the law of sines.

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

    $180^\circ$. The sum of the interior angles in any triangle is always $180^\circ$ due to the properties of parallel lines and transversals.

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

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

12. Find $x$ if vertical angles have measures $(4x-10)^\circ$ and $(2x+30)^\circ$.

    $x=20$. Set $4x - 10 = 2x + 30$ and solve for $x$, since vertical angles are equal.

13. Find $x$ if two angles form a linear pair and have measures $3x^\circ$ and $(2x+30)^\circ$.

    $x=30$. Set up the equation $3x + 2x + 30 = 180$ and solve for $x$, as linear pairs sum to $180^\circ$.

14. What are the side-length ratios of a $45^\circ$-$45^\circ$-$90^\circ$ triangle?

    $1:1:\sqrt{2}$. In a $45^\circ$-$45^\circ$-$90^\circ$ triangle, the legs are equal, and the hypotenuse is leg times $\sqrt{2}$.

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

    $x=70^\circ$. The sum of angles in a triangle is $180^\circ$, so $x = 180^\circ - 50^\circ - 60^\circ$.

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

    $\frac{(n-2)\cdot 180^\circ}{n}$. In a regular $n$-gon, all interior angles are equal, so the total sum is divided by $n$.

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

    It has at least two equal side lengths. An isosceles triangle has at least two sides of equal length by definition.

18. What is the measure of each exterior angle of a regular $n$-gon in degrees?

    $\frac{360^\circ}{n}$. Each exterior angle of a regular $n$-gon is equal, dividing the total sum of $360^\circ$ by $n$.

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

    $360^\circ$. The sum of exterior angles of any convex polygon is $360^\circ$, equivalent to one full rotation.

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

    Corresponding angles are congruent. When parallel lines are cut by a transversal, corresponding angles are equal due to the parallel line postulate.

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

    They are supplementary: sum is $180^\circ$. Angles in a linear pair are adjacent and form a straight line, so their measures add up to $180^\circ$.

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

    They are supplementary: sum is $180^\circ$. Same-side interior angles add up to $180^\circ$ when lines are parallel, forming supplementary pairs.

23. What is the relationship between alternate interior angles for parallel lines cut by a transversal?

    Alternate interior angles are congruent. Alternate interior angles are equal when lines are parallel, as they lie on opposite sides of the transversal inside the parallels.

24. What is the relationship between supplementary angles?

    They sum to $180^\circ$. Supplementary angles are defined as two angles whose measures add up to $180^\circ$.

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