Question

for items 7 - 10, use the figure shown. d is the midpoint of $overline{ab}$. what is $mangle a$? a. 22.62 b. 33.69 c. 45 d. 67.38

Explanation:

Step1: Recall triangle - angle sum property

The sum of angles in a triangle is 180°. In right - triangle ACD, we know that ∠ADC = 90° and we are given part of the information about the angles in the large triangle. First, note that in right - triangle ACD, we can find ∠A using trigonometric relations or angle - sum property. Since D is the mid - point of AB and CD is perpendicular to AB, triangle ABC is isosceles (by the property of the perpendicular bisector of the base of a triangle). Let's use the angle - sum property of a triangle.

Step2: Calculate the sum of non - right angles in right - triangle ACD

In right - triangle ACD, ∠ACD=\frac{134.76^{\circ}}{2}=67.38^{\circ} (because CD is the perpendicular bisector of AB in isosceles triangle ABC).

Step3: Find ∠A

In right - triangle ACD, using the angle - sum property of a triangle (∠A+∠ACD + ∠ADC=180° and ∠ADC = 90°), we have ∠A=180°−90° - 67.38°=22.62°.

Answer:

A. 22.62