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Section I: Multiple Choice (12 questions)

Instructions: Choose the best answer for each question. Show your work where appropriate.

Which of the following best describes a trapezoid?

Select one option

A quadrilateral with exactly one pair of parallel sides.

A quadrilateral with two pairs of parallel sides.

A quadrilateral with four congruent sides.

A quadrilateral with four right angles.

If two parallel lines are intersected by a transversal, and one of the consecutive interior angles measures $70^{\circ}$, what is the measure of the other consecutive interior angle?

Select one option

$20^{\circ}$

$70^{\circ}$

$90^{\circ}$

$110^{\circ}$

Which postulate or theorem can be used to prove that two right triangles are congruent if their hypotenuses and a pair of corresponding legs are congruent?

Select one option

SSS

ASA

AAS

Which of the following is always true for a parallelogram?

Select one option

All four sides are congruent.

The diagonals are congruent.

The diagonals are perpendicular.

The diagonals bisect each other.

A triangle has a base of $15\text{ cm} $and a height of \$ 8\text{ cm}$. What is its area?

Select one option

$23\text{ cm}^2$

$30\text{ cm}^2$

$60\text{ cm}^2$

$120\text{ cm}^2$

What is the volume of a cylinder with a radius of $5\text{ cm} $and a height of \$ 4\text{ cm} $? Use$ \pi$.

Select one option

$20\pi\text{ cm}^3$

$50\pi\text{ cm}^3$

$100\pi\text{ cm}^3$

$400\pi\text{ cm}^3$

What is a chord of a circle?

Select one option

A line that intersects a circle at exactly one point.

A line that intersects a circle at two points and passes through the center.

A line segment connecting two points on a circle.

The distance around the circle.

An inscribed angle in a circle intercepts an arc that measures $50^{\circ}$. What is the measure of the inscribed angle?

Select one option

$25^{\circ}$

$50^{\circ}$

$75^{\circ}$

$100^{\circ}$

10.

In a right triangle, the side opposite angle $A$ is $6\text{ m} $and the hypotenuse is \$ 8\text{ m} $. What is the approximate length of the side adjacent to angle$ A$?

Select one option

$5.29\text{ m}$

$6\text{ m}$

$8\text{ m}$

$10 \text{ m}$

11.

Which of the following is a unique property of a rhombus that is not necessarily true for all parallelograms?

Select one option

Opposite angles are congruent.

Consecutive angles are supplementary.

The diagonals are perpendicular.

Opposite sides are parallel.

12.

Two triangles have corresponding angles that are congruent. These triangles must be:

Select one option

Congruent.

Similar, but not necessarily congruent.

Isosceles.

Equilateral.

13.

A right triangle has legs of length $5 $and \$ 12$. What is the length of the hypotenuse?

Select one option

$7$

$13$

$17$

$\sqrt{119}$

Section II: Free Response (9 questions)

14.

Instructions: Provide detailed solutions or explanations for each question. Show all necessary calculations.

Diagram for Question 15

A semicircle has a diameter of \$10\text{ cm}$.

15.

Calculate the area of the shaded region, which is a semicircle. Round your answer to two decimal places. Use $\pi \approx 3.14159$ .

16.

A spherical ball has a radius of $6\text{ cm} $. Calculate its volume. Express your answer in terms of$ \pi$.

17.

A trapezoid has parallel bases measuring $15\text{ cm} $and \$ 21\text{ cm}$. What is the length of its median?

18.

In parallelogram $ABCD$ , $AB = 2x+5$ and $CD = 3x-5$ . Also, $m\angle A = (4y)^{\circ}$ and $m\angle C = (y+105)^{\circ}$ . Find the values of $x$ and $y$ .

19.

A circle has a radius of $10\text{ cm} $. Find the length of an arc intercepted by a central angle of \$ 90^{\circ} $. Round your answer to two decimal places. Use$ \pi \approx 3.14159$.

20.

From a point on the ground $10\text{ m} $away from the base of a flagpole, the angle of elevation to the top of the flagpole is \$ 60^{\circ}$. Calculate the height of the flagpole to two decimal places.

21.

Point $P$ is outside a circle. A tangent segment $PQ$ is drawn from $P$ to the circle, and the radius at $Q$ is $5\text{ cm} $. If the distance from$ P $to the center of the circle is \$ 13\text{ cm} $, find the length of$ PQ$.

22.

A triangle has vertices at $A(1,1)$ , $B(11,1)$ , and $C(6,7)$ . Calculate the area of the triangle.

23.

Given points $A(1,2)$ and $B(7,10)$ , find the coordinates of the midpoint of segment $AB$ and the distance between $A$ and $B$ .

Section III: Proofs (5 questions)

24.

Instructions: For each problem, write a complete two-column proof. Clearly state your Given, Prove, and provide logical Statements and Reasons.

Proof for Question 25

25.

Given: $\overline{AB} \cong \overline{DE}$ , $\angle A \cong \angle D$ , $\angle B \cong \angle E$ . Prove: $\triangle ABC \cong \triangle DEF$ .

Diagram for Proof Question 26

26.

Given: Line $l$ is parallel to line $m$ . Line $t$ is a transversal. Prove: $\angle 1 \cong \angle 3$ (Alternate Interior Angles are congruent).

Diagram for Proof Question 27

27.

Given: $ABCD$ is a parallelogram. Prove: Opposite sides of a parallelogram are congruent ( $\overline{AB} \cong \overline{CD}$ and $\overline{BC} \cong \overline{DA}$ ).

Diagram for Proof Question 28

28.

Given: $\overline{PQ}$ and $\overline{PR}$ are tangent segments to circle $O$ from external point $P$ . Prove: $\overline{PQ} \cong \overline{PR}$ .

Diagram for Proof Question 29

29.

Given: $\triangle ABC$ is isosceles with $\overline{AB} \cong \overline{AC}$ . $\overline{AD}$ is the angle bisector of $\angle BAC$ . Prove: $\overline{BD} \cong \overline{CD}$ .