Multi-Step Angle Problems

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Multi-Step Angle Problems · Grade 7 · Worksheet 3

1. In the diagram, two parallel lines are cut by a transversal. One of the interior angles on the same side of the transversal is (3x + 15)° and the other interior angle on the same side is (2x + 25)°. What is the value of x? Answer: 28 Solution: Interior angles on the same side of a transversal cutting parallel lines are supplementary, meaning their sum is 180°.
   Full step-by-step solution

   Step 1: Interior angles on the same side of a transversal cutting parallel lines are supplementary, meaning their sum is 180°. Step 2: Set up the equation: (3x + 15) + (2x + 25) = 180 Step 3: Combine like terms: 3x + 2x + 15 + 25 = 180 → 5x + 40 = 180 Step 4: Subtract 40 from both sides: 5x = 140 Step 5: Divide both sides by 5: x = 28 The answer is 28.

2. A triangular park has sides measuring 120 meters, 160 meters, and 200 meters. The city wants to build a similar triangular playground where the longest side is 50 meters. What will be the perimeter of the new playground? Answer: 120 Solution: Identify the scale factor using the longest sides. The original triangle's longest side is 200 m, and the similar triangle's longest side is 50 m. The scale factor is 50 / 200 = 1/4.
   Full step-by-step solution

   Step 1: Identify the scale factor using the longest sides. The original triangle's longest side is 200 m, and the similar triangle's longest side is 50 m. The scale factor is 50 / 200 = 1/4. Step 2: Apply the scale factor to find the other two sides of the similar triangle. The first side becomes 120 * (1/4) = 30 m. The second side becomes 160 * (1/4) = 40 m. Step 3: Calculate the perimeter of the new triangle by adding all its sides: 30 m + 40 m + 50 m = 120 m. The perimeter of the new playground is 120 meters.

3. In triangle ABC, angle A is 3x + 15, angle B is 2x - 5, and angle C is x + 20. Find the measure of each angle. Answer: Angle A = 105°, Angle B = 55°, Angle C = 20° Solution: Write the sum of the angles equal to 180. (3x + 15) + (2x - 5) + (x + 20) = 180 Combine like terms. 3x + 2x + x = 6x 15 - 5 + 20 = 30 So, 6x + 30 = 180 Solve for x.
   Full step-by-step solution

   Step 1: Write the sum of the angles equal to 180. (3x + 15) + (2x - 5) + (x + 20) = 180 Step 2: Combine like terms. 3x + 2x + x = 6x 15 - 5 + 20 = 30 So, 6x + 30 = 180 Step 3: Solve for x. 6x + 30 = 180 6x = 180 - 30 6x = 150 x = 150 ÷ 6 x = 25 Step 4: Find each angle. Angle A = 3(25) + 15 = 75 + 15 = 90° Angle B = 2(25) - 5 = 50 - 5 = 45° Angle C = 25 + 20 = 45° Step 5: Check the sum. 90 + 45 + 45 = 180° Final answer: Angle A = 90°, Angle B = 45°, Angle C = 45°

4. Charlotte is designing a triangular window for a modern art studio. The window is a right triangle where one acute angle is 27 degrees. A decorative metal strip runs along the longest side of the window, which is 62 inches. What is the measure of the other acute angle in the triangle? Answer: 63 Solution: A right triangle has one angle that measures 90 degrees. The sum of all angles in a triangle is 180 degrees. Let the unknown acute angle be x.
   Full step-by-step solution

   Step 1: A right triangle has one angle that measures 90 degrees. Step 2: The sum of all angles in a triangle is 180 degrees. Step 3: Let the unknown acute angle be x. Step 4: Write the equation: 90 + 27 + x = 180 Step 5: Combine the known angles: 117 + x = 180 Step 6: Subtract 117 from both sides: x = 180 - 117 = 63 Step 7: The other acute angle measures 63 degrees. The answer is 63.

5. Emma is building a triangular wooden frame for a kite. Two of the angles in the triangle measure 35° and 85°. She needs to cut the third angle and also wants to check that the sum of the largest angle and the smallest angle equals twice the middle angle. Is this true for her triangle? What is the measure of the third angle? Answer: 60° Solution: Find the third angle. The sum of angles in a triangle is 180°. So, 35° + 85° + x = 180°.
   Full step-by-step solution

   Step 1: Find the third angle. The sum of angles in a triangle is 180°. So, 35° + 85° + x = 180°. Then 120° + x = 180°, so x = 60°. Step 2: The largest angle is 85°, the smallest is 35°, and the middle is 60°. Check if largest + smallest = 2 × middle: 85° + 35° = 120°, and 2 × 60° = 120°. Yes, it is true. The third angle is 60°.

6. (3/4 × 2/3) ÷ (1/2 + 1/6) = ? Answer: 3/4 Solution: We have: (3/4 × 2/3) ÷ (1/2 + 1/6) Simplify the numerator (3/4 × 2/3) Multiply the numerators: 3 × 2 = 6 Multiply the denominators: 4 × 3 = 12 So 3/4 × 2/3 = 6/12 Simplify 6/12: divide numerator and denominator by 6 → 1/2 So numerator = 1/2 Simplify the denominator (1/2 + 1/6) Find a common…
   Full step-by-step solution

   Let's solve step-by-step. We have: (3/4 × 2/3) ÷ (1/2 + 1/6) --- **Step 1: Simplify the numerator (3/4 × 2/3)** Multiply the numerators: 3 × 2 = 6 Multiply the denominators: 4 × 3 = 12 So 3/4 × 2/3 = 6/12 Simplify 6/12: divide numerator and denominator by 6 → 1/2 So numerator = 1/2 --- **Step 2: Simplify the denominator (1/2 + 1/6)** Find a common denominator for 1/2 and 1/6. The least common denominator of 2 and 6 is 6. 1/2 = 3/6 1/6 = 1/6 Add: 3/6 + 1/6 = 4/6 Simplify 4/6: divide numerator and denominator by 2 → 2/3 So denominator = 2/3 --- **Step 3: Perform the division (1/2) ÷ (2/3)** Dividing by a fraction is the same as multiplying by its reciprocal: (1/2) ÷ (2/3) = (1/2) × (3/2) Multiply numerators: 1 × 3 = 3 Multiply denominators: 2 × 2 = 4 Result = 3/4 --- **Final Answer:** 3/4

7. Emma is designing a triangular patio with sides measuring 18 feet, 24 feet, and 30 feet. She wants to create a similar triangular flower bed where the shortest side is only 12 feet. What will be the perimeter of the smaller flower bed? Answer: 48 Solution: Identify the shortest side of the original triangle: 18 feet Find the scale factor between the triangles: smaller shortest side / original shortest side = 12 / 18 = 2/3 Apply the scale factor to find all sides of the smaller triangle: - First side: 18 × (2/3) = 12 feet - Second side: 24 × (2/3)…
   Full step-by-step solution

   Step 1: Identify the shortest side of the original triangle: 18 feet Step 2: Find the scale factor between the triangles: smaller shortest side / original shortest side = 12 / 18 = 2/3 Step 3: Apply the scale factor to find all sides of the smaller triangle: - First side: 18 × (2/3) = 12 feet - Second side: 24 × (2/3) = 16 feet - Third side: 30 × (2/3) = 20 feet Step 4: Calculate the perimeter of the smaller triangle: 12 + 16 + 20 = 48 feet The answer is 48.

8. Two supplementary angles have measures (3x + 17)° and (5x - 9)°. Find the value of x. Answer: 21.5 Solution: Since the angles are supplementary, their sum is 180°. (3x + 17) + (5x - 9) = 180 Combine like terms. 3x + 5x + 17 - 9 = 180 8x + 8 = 180 Subtract 8 from both sides.
   Full step-by-step solution

   Step 1: Since the angles are supplementary, their sum is 180°. (3x + 17) + (5x - 9) = 180 Step 2: Combine like terms. 3x + 5x + 17 - 9 = 180 8x + 8 = 180 Step 3: Subtract 8 from both sides. 8x = 172 Step 4: Divide both sides by 8. x = 172 ÷ 8 x = 21.5 The answer is 21.5.