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This exam was created by a large language model. Please use it with caution, as inaccuracies may be present. Always verify critical information.

What is a statement that is accepted as true without proof?

Select one option

Theorem

Postulate

Conjecture

Lemma

Which property is NOT always true for a parallelogram?

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Opposite sides are parallel

Opposite angles are congruent

Diagonals are congruent

Consecutive angles are supplementary

Given $\triangle ABC$ and $\triangle DEF$ . If $AB=DE$ , $\angle B = \angle E$ , and $BC=EF$ , which postulate proves $\triangle ABC \cong \triangle DEF$ ?

Select one option

SSS

SAS

ASA

AAS

What is the area of a triangle with a base of $10$ cm and a height of $8$ cm?

Select one option

80 cm $^2$

40 cm $^2$

18 cm $^2$

36 cm $^2$

A cylinder has a radius of $3$ cm and a height of $5$ cm. What is its volume? (Use $\pi \approx 3.14$ )

Select one option

45 $\pi$ cm $^3$

30 $\pi$ cm $^3$

15 $\pi$ cm $^3$

75 $\pi$ cm $^3$

A line that intersects a circle at exactly one point is called a:

Select one option

Chord

Secant

Tangent

Radius

In a right-angled triangle, if $\sin A = \frac{\text{opposite}}{\text{hypotenuse}}$ , then $\cos A$ is equal to:

Select one option

$\frac{\text{opposite}}{\text{adjacent}}$

$\frac{\text{adjacent}}{\text{hypotenuse}}$

$\frac{\text{hypotenuse}}{\text{opposite}}$

$\frac{\text{hypotenuse}}{\text{adjacent}}$

In a $45^\circ-45^\circ-90^\circ$ right triangle, if one leg has a length of $5$ , what is the length of the hypotenuse?

Select one option

$5\sqrt{2}$

$10$

$5\sqrt{3}$

$2.5$

Which quadrilateral always has diagonals that are perpendicular bisectors of each other?

Select one option

Parallelogram

Rectangle

Rhombus

Trapezoid

10.

An angle whose vertex is on a circle and whose sides are chords of the circle is called an:

Select one option

Central angle

Inscribed angle

Circumscribed angle

Tangent angle

11.

When looking up at an object, the angle between the horizontal line of sight and the line of sight to the object is called the:

Select one option

Angle of depression

Angle of incidence

Angle of reflection

Angle of elevation

12.

Which of the following is NOT a valid congruence postulate for triangles?

Select one option

AAA

SSS

ASA

13.

The sum of the interior angles of any quadrilateral is:

Select one option

$180^\circ$

$270^\circ$

$360^\circ$

$540^\circ$

14.

A circle has a diameter of $14$ cm. What is its area? (Use $\pi \approx 3.14$ )

Select one option

28 $\pi$ cm $^2$

49 $\pi$ cm $^2$

196 $\pi$ cm $^2$

14 $\pi$ cm $^2$

15.

What is the formula for the volume of a sphere with radius $r$ ?

Select one option

$\frac{1}{3}\pi r^2 h$

$\pi r^2 h$

$4\pi r^2$

$\frac{4}{3}\pi r^3$

16.

A composite shape is formed by a rectangle with dimensions $10 \text{ cm} \times 6 \text{ cm}$ and a semicircle attached to one of the $6 \text{ cm}$ sides. Calculate the total area of the shape. Use $\pi \approx 3.14$ .

17.

A cone has a height of $12$ inches and a radius of $5$ inches. What is its volume? (Use $\pi \approx 3.14$ )

18.

A ladder $15$ feet long leans against a wall. The base of the ladder is $7$ feet from the wall. What is the angle of elevation of the ladder to the nearest degree?

19.

In parallelogram $ABCD$ , $AB = 3x + 5$ and $CD = x + 15$ . Also, $\angle A = 4y - 10$ and $\angle B = 2y + 40$ . Find the values of $x$ and $y$ .

20.

In circle $O$ , chord $AB$ is $16$ cm long and is $6$ cm from the center. Find the radius of the circle.

21.

Given $\triangle PQR$ and $\triangle XYZ$ . If $PQ = XY$ , $\angle P = \angle X$ , and $\angle Q = \angle Y$ , and $QR = 12$ units, what is the length of $YZ$ ? Justify your answer.

22.

A triangle has vertices at $A(1, 2)$ , $B(4, 2)$ , and $C(1, 6)$ . a) Find the lengths of all three sides. b) Determine if the triangle is a right-angled triangle. Justify your answer.

23.

Given: $AE \cong CE$ and $BE \cong DE$ . Prove: $\triangle ABE \cong \triangle CDE$ .

24.

Prove that the opposite sides of a parallelogram are congruent.

25.

Given: $\angle ABD$ and $\angle DBC$ form a linear pair. Prove: $\angle ABD$ and $\angle DBC$ are supplementary.