Question

use efficient methods find each measure. 20. ( m angle c a d ) 21. ( m angle a c d ) 22. ( m angle a c b ) 23. ( m angle a b c )

Explanation:

Step 1: Find \(m\angle ACD\)

Since \(\angle ACD\) and \(\angle ADC\) are related. We know that \(\angle ADC = 92^{\circ}\) and \(AD = AC\) (marked as equal). In \(\triangle ADC\), using the property that the sum of angles in a triangle is \(180^{\circ}\). Let \(m\angle CAD=m\angle ACD=x\) (isosceles triangle property: angles opposite equal sides are equal). Then \(x + x+92^{\circ}=180^{\circ}\), \(2x=180^{\circ}- 92^{\circ}=88^{\circ}\), \(x = 44^{\circ}\). So \(m\angle ACD = 44^{\circ}\) and \(m\angle CAD=44^{\circ}\)

Step 2: Find \(m\angle ACB\)

\(\angle ACB\) and \(\angle ACD\) are supplementary (linear - pair). So \(m\angle ACB=180^{\circ}-m\angle ACD\). Since \(m\angle ACD = 44^{\circ}\), then \(m\angle ACB = 136^{\circ}\)

Step 3: Find \(m\angle ABC\)

In \(\triangle ABC\), assume \(AC = AB\) (marked as equal). Let \(m\angle ABC=y\). Using the angle - sum property of a triangle in \(\triangle ABC\), \(m\angle BAC + m\angle ABC+m\angle ACB=180^{\circ}\). Since \(m\angle BAC=m\angle CAD = 44^{\circ}\) and \(m\angle ACB = 136^{\circ}\), then \(44^{\circ}+y + 136^{\circ}=180^{\circ}\), \(y+180^{\circ}=180^{\circ}\), \(y = 22^{\circ}\)

Answer:

1. \(m\angle CAD = 44^{\circ}\)
2. \(m\angle ACD = 44^{\circ}\)
3. \(m\angle ACB = 136^{\circ}\)
4. \(m\angle ABC = 22^{\circ}\)