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If two parallel lines are intersected by a transversal, which of the following pairs of angles are always equal?

Select one option

Alternate interior angles

Consecutive interior angles

Corresponding angles

Supplementary angles

In $\triangle ABC$ , if $\angle A = 50^\circ$ and $\angle B = 70^\circ$ , what is the measure of $\angle C$ ?

Select one option

$50^\circ$

$60^\circ$

$70^\circ$

$80^\circ$

Which of the following is NOT a valid criterion for proving triangle congruence?

Select one option

SSS

SSA

ASA

AAS

If two triangles are similar and the ratio of their corresponding sides is $2:3$ , what is the ratio of their areas?

Select one option

$2:3$

$4:9$

$8:27$

$1:1$

Which property is unique to a rhombus among parallelograms?

Select one option

Diagonals bisect each other

Opposite angles are equal

Diagonals are perpendicular

Opposite sides are parallel

In an isosceles trapezoid, which of the following statements is true?

Select one option

Diagonals are perpendicular

All sides are equal

Non-parallel sides are equal

Opposite angles are supplementary

What is the area of a circle with a radius of $7 \text{ cm}$ ? Use $\pi \approx \frac{22}{7}$ .

Select one option

$22 \text{ cm}^2$

$44 \text{ cm}^2$

$154 \text{ cm}^2$

$112 \text{ cm}^2$

The volume of a cylinder with radius $r$ and height $h$ is given by:

Select one option

$\pi r^2 h$

$2\pi rh$

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

$2\pi r^2 + 2\pi rh$

An angle inscribed in a semicircle measures:

Select one option

$45^\circ$

$60^\circ$

$90^\circ$

$180^\circ$

10.

A line tangent to a circle is perpendicular to the radius drawn to the point of tangency. This angle is always:

Select one option

$0^\circ$

$45^\circ$

$90^\circ$

$180^\circ$

11.

In a right-angled triangle, if the opposite side to an angle $\theta$ is $6$ units and the hypotenuse is $10$ units, what is $\sin \theta$ ?

Select one option

$0.6$

$0.8$

$1.25$

$0.75$

12.

What is the exact value of $\tan 45^\circ$ ?

Select one option

$0$

$1$

$\frac{\sqrt{2}}{2}$

$\frac{\sqrt{3}}{2}$

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      <line x1="10" y1="20" x2="100" y2="130" stroke="black" stroke-width="1"/>

      <circle cx="100" cy="130" r="2" fill="black"/>
      <text x="95" y="145" font-size="12">E</text>

      <path d="M185 20 L190 20 L190 25" fill="none" stroke="black"/>
      <text x="170" y="30" font-size="12">130°</text>

      <path d="M105 130 L100 130 L100 125" fill="none" stroke="black"/>
      <text x="110" y="120" font-size="12">80°</text>
    </svg>

13.

In the diagram below, line $AB$ is parallel to line $CD$ . If $\angle ABE = 130^\circ$ and $\angle CDE = 80^\circ$ , find the measure of $\angle BED$ . Show your steps.

      <line x1="40" y1="75" x2="160" y2="75" stroke="black" stroke-width="1"/>
      <text x="30" y="70" font-size="12">D</text>
      <text x="165" y="70" font-size="12">E</text>

      <text x="50" y="45" font-size="10">4</text> <!-- AD -->
      <text x="50" y="110" font-size="10">6</text> <!-- DB -->
      <text x="140" y="45" font-size="10">5</text> <!-- AE -->
      <text x="140" y="110" font-size="10">x</text> <!-- EC -->
    </svg>

14.

In $\triangle ABC$ , $DE \parallel BC$ . If $AD = 4 \text{ cm}$ , $DB = 6 \text{ cm}$ , and $AE = 5 \text{ cm}$ , find the length of $EC$ . Show your work.

15.

The vertices of a parallelogram $ABCD$ are $A(1, 2)$ , $B(5, 2)$ , $C(7, 6)$ . Find the coordinates of vertex $D$ . Show your method.

      <path d="M 20 60 A 80 80 0 0 1 180 60" fill="none" stroke="black" stroke-width="1"/>
    </svg>

16.

A design consists of a rectangle with dimensions $8 \text{ m}$ by $6 \text{ m}$ with a semicircle attached to one of its $8 \text{ m}$ sides. Calculate the total area of the design. Use $\pi = 3.14$ .

17.

A solid toy is in the form of a hemisphere surmounted by a cone. The radius of the hemisphere is $3.5 \text{ cm}$ and the total height of the toy is $9.5 \text{ cm}$ . Find the volume of the toy. (Use $\pi = \frac{22}{7}$ )

18.

A ladder $10 \text{ m}$ long reaches a window $8 \text{ m}$ above the ground. Find the angle of elevation of the ladder to the nearest degree. (You may use a calculator for trigonometric values).

D E

      <line x1="20" y1="20" x2="180" y2="130" stroke="black" stroke-width="1"/>
      <!-- Point C is the midpoint of AE based on AC=CE -->
      <circle cx="100" cy="75" r="2" fill="black"/>
      <text x="95" y="70" font-size="12">C</text>

      <line x1="80" y1="20" x2="120" y2="130" stroke="black" stroke-width="1"/>
    </svg>

19.

Given: $AB \parallel DE$ , and $AC = CE$ . Prove: $\triangle ABC \cong \triangle EDC$ . Provide a formal two-column proof (Statements and Reasons).

          <line x1="10" y1="110" x2="190" y2="110" stroke="black" stroke-width="1"/>
          <text x="5" y="105" font-size="12">C</text>
          <text x="195" y="105" font-size="12">D</text>

          <line x1="50" y1="10" x2="150" y2="140" stroke="black" stroke-width="1"/>
          <text x="45" y="5" font-size="12">E</text>
          <text x="155" y="145" font-size="12">F</text>

          <circle cx="80" cy="40" r="2" fill="black"/>
          <text x="75" y="30" font-size="12">G</text>
          <circle cx="120" cy="110" r="2" fill="black"/>
          <text x="115" y="100" font-size="12">H</text>

          <path d="M85 40 L80 40 L80 45" fill="none" stroke="black"/>
          <text x="90" y="50" font-size="12">1</text>

          <path d="M125 110 L120 110 L120 105" fill="none" stroke="black"/>
          <text x="130" y="100" font-size="12">2</text>
        </svg>

20.

Given: Line $AB$ and $CD$ are intersected by transversal $EF$ at $G$ and $H$ respectively. $\angle AGH = \angle GHD$ . Prove: $AB \parallel CD$ . Provide a formal two-column proof (Statements and Reasons).

      <line x1="20" y1="75" x2="180" y2="75" stroke="black" stroke-width="1"/>
      <line x1="100" y1="20" x2="100" y2="130" stroke="black" stroke-width="1"/>

      <circle cx="100" cy="75" r="2" fill="black"/>
      <text x="105" y="70" font-size="12">O</text>
    </svg>

21.

Given: Quadrilateral $ABCD$ where diagonals $AC$ and $BD$ bisect each other at $O$ . Prove: $ABCD$ is a parallelogram. Provide a formal two-column proof (Statements and Reasons).