Questions: Complete the following proof: Given: EH→ bisects ∠DEG Prove: m∠1+m∠3=m∠HEF

Transcript text: 7. Complete the following proof: Glven: $\overrightarrow{\mathrm{EH}}$ bisects $\angle \mathrm{DEG}$ Prove: $m \angle 1+m \angle 3=m \angle H E F$

Solution

Solution Steps

To solve this problem, we need to use the property of angle bisectors. Since $\overrightarrow{\mathrm{EH}}$ bisects $\angle \mathrm{DEG}$, it divides the angle into two equal parts. We can express the measures of these angles in terms of $m \angle 1$ and $m \angle 3$, and then relate them to $m \angle H E F$.

Step 1: Given Information

We are given that $\overrightarrow{\mathrm{EH}}$ bisects $\angle \mathrm{DEG}$. This means that:

\[ m \angle 1 = m \angle 3 = \frac{1}{2} m \angle DEG \]

Step 2: Calculate the Angles

Assuming $m \angle DEG = 100^\circ$, we can find the measures of angles 1 and 3:

\[ m \angle 1 = m \angle 3 = \frac{100^\circ}{2} = 50^\circ \]

Step 3: Relate to $m \angle H E F$

Since $\overrightarrow{\mathrm{EH}}$ bisects $\angle \mathrm{DEG}$, we can express $m \angle H E F$ as the sum of $m \angle 1$ and $m \angle 3$:

\[ m \angle H E F = m \angle 1 + m \angle 3 = 50^\circ + 50^\circ = 100^\circ \]