Question

15 a ramp is designed with the profile of a right triangle. the measure of one acute angle is 2 times the measur of the other acute angle. find the measure of the two acute angles. a 45°, 45° b 11.25°, 78.75° c 22.5°, 67.5° d 30°, 60° 16. find the measure of each acute angle. 17. find the value of the measure of angle a. a 13 b 60 c 26 d 13√3

Explanation:

Step1: Recall the property of right - triangle angles

In a right - triangle, the sum of the two acute angles is \(90^{\circ}\). Let one acute angle be \(x\), then the other acute angle is \(2x\). So, \(x + 2x=90^{\circ}\).

Step2: Solve the equation for \(x\)

Combine like terms: \(3x = 90^{\circ}\). Divide both sides by 3: \(x=\frac{90^{\circ}}{3}=30^{\circ}\). Then \(2x = 2\times30^{\circ}=60^{\circ}\).

Step1: Use the property of right - triangle angles

In a right - triangle, the sum of the two non - right angles is \(90^{\circ}\). So, \((2x + 6)+(5x-14)=90\).

Step2: Simplify the left - hand side of the equation

Combine like terms: \(2x+5x+6 - 14=90\), which gives \(7x-8 = 90\).

Step3: Solve for \(x\)

Add 8 to both sides: \(7x=90 + 8=98\). Divide both sides by 7: \(x = 14\).

Step4: Find the measure of each acute angle

For the first acute angle: \(2x+6=2\times14 + 6=28+6=34^{\circ}\). For the second acute angle: \(5x-14=5\times14-14=(5 - 1)\times14=56^{\circ}\).

Step1: Use the angle - sum property of a triangle

The sum of the interior angles of a triangle is \(180^{\circ}\). Given two angles \(B = 60^{\circ}\) and \(C=60^{\circ}\). Let \(\angle A=x\). Then \(x+60^{\circ}+60^{\circ}=180^{\circ}\).

Step2: Solve for \(x\)

Combine like terms: \(x+120^{\circ}=180^{\circ}\). Subtract \(120^{\circ}\) from both sides: \(x=180^{\circ}-120^{\circ}=60^{\circ}\).

Answer:

d. \(30^{\circ},60^{\circ}\)