Question

3-72. use what you know about the angles of a triangle to determine the value of x and the angle(s) in each triangle below. homework help
a.
b.
c.
d.
3-74. solve for the missing lengths in the sets of similar figures below. homework help
a. △abc∼△opq

Explanation:

Step1: Recall angle - sum property of a triangle

The sum of the interior angles of a triangle is 180°. A right - angled triangle has one angle equal to 90°.

Step2: Solve for part a

We know one angle is 90° and another is 79°. Let the third angle be \(x\). Using the angle - sum property \(90 + 79+x=180\). Then \(x = 180-(90 + 79)=180 - 169 = 11^{\circ}\).

Step3: Solve for part b

The triangle is a right - angled isosceles triangle. Let the two equal angles be \(x\). Then \(90+x + x=180\), which simplifies to \(90 + 2x=180\). Subtract 90 from both sides: \(2x=180 - 90=90\). Divide both sides by 2: \(x = 45^{\circ}\).

Step4: Solve for part c

The triangle is right - angled. The angles are 90°, \(x\), and \(2x\). Using the angle - sum property \(90+x + 2x=180\), which simplifies to \(90+3x = 180\). Subtract 90 from both sides: \(3x=180 - 90 = 90\). Divide both sides by 3: \(x = 30^{\circ}\).

Step5: Solve for part d

The triangle is right - angled. One angle is 90° and another is 22°. Let the third angle be \(x\). Using the angle - sum property \(90+22 + x=180\). Then \(x=180-(90 + 22)=180 - 112 = 68^{\circ}\).

Answer:

a. \(x = 11^{\circ}\)
b. \(x = 45^{\circ}\)
c. \(x = 30^{\circ}\)
d. \(x = 68^{\circ}\)