Question

1. abcd is a rhombus.

find the measure of ∠edc.
find the measure of ∠dcb.
find the measure of ∠bae.

Explanation:

Step1: Properties of a rhombus

In a rhombus, the diagonals bisect the angles. So, \( \angle EDC=\angle EDA = 72^{\circ}\)

Step2: Find \(\angle ADC\)

\(\angle ADC=\angle EDC+\angle EDA\)
\(\angle ADC = 72^{\circ}+72^{\circ}=144^{\circ}\)

Step3: Use adjacent - angles property of rhombus

In a rhombus \(ABCD\), \(AD\parallel BC\), so \(\angle ADC+\angle DCB = 180^{\circ}\) (co - interior angles)
\(\angle DCB=180^{\circ}-\angle ADC\)
Substitute \(\angle ADC = 144^{\circ}\)
\(\angle DCB = 180^{\circ}-144^{\circ}=36^{\circ}\)

Step4: Use angle - bisecting property of diagonal

Since \(AC\) is a diagonal of the rhombus and bisects \(\angle DAB\) and \(\angle DCB\)
\(\angle BAE=\frac{1}{2}\angle DAB\)
Since \(AD\parallel BC\), \(\angle DAB+\angle ABC = 180^{\circ}\) and \(\angle ABC=\angle ADC = 144^{\circ}\) (opposite angles of a rhombus are equal)
\(\angle DAB = 36^{\circ}\)
\(\angle BAE=\frac{1}{2}\times36^{\circ}=18^{\circ}\)

Answer:

\(\angle EDC = 72^{\circ}\), \(\angle DCB=36^{\circ}\), \(\angle BAE = 18^{\circ}\)