Multi-Step Angle Problems

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Multi-Step Angle Problems · Grade 7 · Worksheet 3

1. Hana is designing a triangular sail for her model boat. She draws a diagram of a right triangle where one acute angle is twice the measure of the other acute angle. The smallest angle of the triangle is labeled x. Find the measure of each angle in the triangle. Answer: 30, 60, 90 Solution: Let the smallest acute angle be x. Since one acute angle is twice the other, the other acute angle is 2x. The triangle is a right triangle, so the right angle is 90 degrees.
   Full step-by-step solution

   Step 1: Let the smallest acute angle be x. Since one acute angle is twice the other, the other acute angle is 2x. The triangle is a right triangle, so the right angle is 90 degrees. Step 2: The sum of the angles in any triangle is 180 degrees. Write the equation: x + 2x + 90 = 180. Step 3: Combine like terms: 3x + 90 = 180. Step 4: Subtract 90 from both sides: 3x = 90. Step 5: Divide both sides by 3: x = 30. Step 6: The smallest angle is x = 30 degrees. The other acute angle is 2x = 60 degrees. The right angle is 90 degrees. Step 7: Check: 30 + 60 + 90 = 180. Correct. The three angles are 30, 60, and 90 degrees.

2. 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.

3. In triangle ABC, angle A = 2x, angle B = 3x, and angle C = 4x. Find the value of x and the measure of each angle. Answer: x = 20, angle A = 40°, angle B = 60°, angle C = 80° Solution: Write the equation for the sum of angles in a triangle. Angle A + Angle B + Angle C = 180 2x + 3x + 4x = 180 Combine like terms. 9x = 180 Solve for x.
   Full step-by-step solution

   Step 1: Write the equation for the sum of angles in a triangle. Angle A + Angle B + Angle C = 180 2x + 3x + 4x = 180 Step 2: Combine like terms. 9x = 180 Step 3: Solve for x. x = 180 ÷ 9 x = 20 Step 4: Find each angle. Angle A = 2x = 2 × 20 = 40° Angle B = 3x = 3 × 20 = 60° Angle C = 4x = 4 × 20 = 80° Step 5: Check the sum. 40° + 60° + 80° = 180° ✓ Final answer: x = 20, angle A = 40°, angle B = 60°, angle C = 80°

4. Tane is building a model of a city skyline. He has two parallel streets represented by lines on his baseboard. A third street (a transversal) crosses both parallel streets. One of the angles formed between the transversal and the first parallel street measures (4x + 18) degrees. The corresponding angle on the second parallel street measures (7x - 42) degrees. Tane needs to know the value of x to correctly position his buildings. What is the value of x? Answer: 20 Solution: Recall that corresponding angles formed by a transversal crossing two parallel lines are congruent (equal in measure). Set up the equation: 4x + 18 = 7x - 42 Subtract 4x from both sides: 18 = 3x - 42 Add 42 to both sides: 60 = 3x Divide both sides by 3: x = 20 Check: 4(20) + 18 = 80 + 18 = 98…
   Full step-by-step solution

   Step 1: Recall that corresponding angles formed by a transversal crossing two parallel lines are congruent (equal in measure). Step 2: Set up the equation: 4x + 18 = 7x - 42 Step 3: Subtract 4x from both sides: 18 = 3x - 42 Step 4: Add 42 to both sides: 60 = 3x Step 5: Divide both sides by 3: x = 20 Step 6: Check: 4(20) + 18 = 80 + 18 = 98 degrees; 7(20) - 42 = 140 - 42 = 98 degrees. They match. The value of x is 20.

5. (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

6. Emma is designing a triangular patio for her backyard. The angles of the triangle form a geometric sequence where the smallest angle is 20 degrees and the common ratio is 2. What is the measure of the largest angle in her triangular patio design? Answer: 80 Solution: Let the three angles be a, ar, and ar², where a is the smallest angle and r is the common ratio. We know a = 20 and r = 2. Check that these angles sum to 180: 20 + 40 + 80 = 140 + 40 = 180.
   Full step-by-step solution

   Step 1: Let the three angles be a, ar, and ar², where a is the smallest angle and r is the common ratio. Step 2: We know a = 20 and r = 2. Step 3: So the angles are: 20, 20×2 = 40, and 20×2² = 20×4 = 80. Step 4: Check that these angles sum to 180: 20 + 40 + 80 = 140 + 40 = 180. Step 5: The largest angle is 80 degrees. The answer is 80.

7. Olivia is building a model of a wooden bridge for her science project. The bridge's main support beam is designed as a triangle. Two angles of the triangle measure 45 degrees and 85 degrees. The third angle is divided into two smaller angles by a support wire. One of these smaller angles is 20 degrees more than the other. What are the measures of the two smaller angles? Answer: 15 degrees and 35 degrees Solution: Find the measure of the third angle in the triangle. The sum of angles in a triangle is 180 degrees. Third angle = 180 - 45 - 85 = 50 degrees.
   Full step-by-step solution

   Step 1: Find the measure of the third angle in the triangle. The sum of angles in a triangle is 180 degrees. Third angle = 180 - 45 - 85 = 50 degrees. Step 2: Let the smaller of the two angles be x degrees. The larger angle is 20 degrees more, so it is x + 20 degrees. Step 3: The sum of these two angles equals the third angle from the triangle. x + (x + 20) = 50 2x + 20 = 50 2x = 30 x = 15 Step 4: Find the measures of both angles. Smaller angle = x = 15 degrees Larger angle = x + 20 = 15 + 20 = 35 degrees The two smaller angles are 15 degrees and 35 degrees.

8. Two supplementary angles have measures (3x + 15)° and (2x + 35)°. Find the value of x. Answer: 26 Solution: Since the angles are supplementary, their sum is 180°. (3x + 15) + (2x + 35) = 180 Combine like terms. 3x + 2x = 5x 15 + 35 = 50 So, 5x + 50 = 180 Subtract 50 from both sides.
   Full step-by-step solution

   Step 1: Since the angles are supplementary, their sum is 180°. (3x + 15) + (2x + 35) = 180 Step 2: Combine like terms. 3x + 2x = 5x 15 + 35 = 50 So, 5x + 50 = 180 Step 3: Subtract 50 from both sides. 5x = 130 Step 4: Divide both sides by 5. x = 26 The answer is 26.