Geometry Worksheet for 15-Year-Olds — High School Math Practice

Free worksheet covering Math for age 15 students. Created with Learning Corner's AI-powered tools.

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Instructions

Solve the following geometry problems, showing your work in the space provided. Use π ≈ 3.14 for your calculations. Round final answers to two decimal places where necessary.

Section A: Angle Relationships

  1. Two straight lines, Line A and Line B, intersect at point P. The angle formed, ∠1, measures 65°. Find the measures of the other three angles (∠2, ∠3, and ∠4) created by the intersection.
    • ∠2 is adjacent to ∠1 (forms a straight line).
    • ∠3 is vertically opposite to ∠1.
    • ∠4 is vertically opposite to ∠2.

Section B: The Pythagorean Theorem

  1. A right-angled triangle has two shorter sides (legs) measuring 9 cm and 12 cm. What is the length of the longest side (the hypotenuse)?
  2. You place a 17-foot ladder against a wall. The base of the ladder is 8 feet from the base of the wall. How high up the wall does the ladder reach?

Section C: Circles

  1. A circular garden has a diameter of 20 meters.
    a) What is the circumference of the garden?
    b) What is the area of the garden?

Section D: Volume and Surface Area

  1. A cylindrical water tank has a radius of 4 feet and a height of 10 feet.
    a) Calculate the volume of the tank. (Volume = πr2h)
    b) Calculate the total surface area of the tank (including the top and bottom). (Surface Area = 2πrh + 2πr2)

Section E: Properties of Quadrilaterals

  1. In parallelogram ABCD, the measure of angle A is (2x + 25)° and the measure of angle B is (3x + 15)°. Find the value of x, and then determine the measure of all four angles (∠A, ∠B, ∠C, and ∠D). Remember the properties of angles in a parallelogram.

Answer Key

1. Angle Relationships

  • ∠3: Vertically opposite angles are equal. So, ∠3 = ∠1 = 65°.
  • ∠2: Angles on a straight line add up to 180°. So, ∠2 = 180° - ∠1 = 180° - 65° = 115°.
  • ∠4: Vertically opposite to ∠2. So, ∠4 = ∠2 = 115°.

2. Pythagorean Theorem (Hypotenuse)

Using the formula a2 + b2 = c2:
92 + 122 = c2
81 + 144 = c2
225 = c2
c = √225 = 15 cm.
The hypotenuse is 15 cm long.

3. Pythagorean Theorem (Leg)

The ladder is the hypotenuse (c = 17 ft), the distance from the wall is a leg (a = 8 ft). We need to find the other leg (b).
a2 + b2 = c2
82 + b2 = 172
64 + b2 = 289
b2 = 289 - 64
b2 = 225
b = √225 = 15 ft.
The ladder reaches 15 feet up the wall.

4. Circles

The diameter is 20 m, so the radius (r) is 10 m.
a) Circumference = 2πr = 2 * 3.14 * 10 = 62.8 meters.
b) Area = πr2 = 3.14 * (10)2 = 3.14 * 100 = 314 square meters.

5. Volume and Surface Area

Radius (r) = 4 ft, Height (h) = 10 ft.
a) Volume = πr2h = 3.14 * (4)2 * 10 = 3.14 * 16 * 10 = 502.4 cubic feet.
b) Surface Area = 2πrh + 2πr2 = (2 * 3.14 * 4 * 10) + (2 * 3.14 * 42)
= (251.2) + (2 * 3.14 * 16) = 251.2 + 100.48 = 351.68 square feet.

6. Properties of Quadrilaterals

In a parallelogram, consecutive angles are supplementary (add up to 180°).
(2x + 25) + (3x + 15) = 180
5x + 40 = 180
5x = 140
x = 28.

Now find the angles:
∠A = 2(28) + 25 = 56 + 25 = 81°
∠B = 3(28) + 15 = 84 + 15 = 99°
Opposite angles are equal, so:
∠C = ∠A = 81°
∠D = ∠B = 99°
(Check: 81 + 99 + 81 + 99 = 360°)
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