Question

1. given: mt and ha intersect at b, ma||ht, and mt bisects ha. prove: ma≅ht

Explanation:

Step1: Use the property of parallel - lines and angle - bisector

Since $\overline{MA}\parallel\overline{HT}$ and $\overline{MT}$ bisects $\overline{HA}$ at point $B$, we know that $\angle MBA=\angle BHT$ (alternate - interior angles) and $\angle AMB=\angle MTH$ (alternate - interior angles). Also, because $\overline{MT}$ bisects $\overline{HA}$, we have $HB = BA$.

Step2: Prove triangle congruence

In $\triangle MAB$ and $\triangle THB$, we have:

• $\angle MBA=\angle BHT$ (alternate - interior angles from $\overline{MA}\parallel\overline{HT}$)
• $HB = BA$ (given that $\overline{MT}$ bisects $\overline{HA}$)
• $\angle AMB=\angle MTH$ (alternate - interior angles from $\overline{MA}\parallel\overline{HT}$)

By the Angle - Side - Angle (ASA) congruence criterion, $\triangle MAB\cong\triangle THB$.

Step3: Use the property of congruent triangles

Since $\triangle MAB\cong\triangle THB$, the corresponding sides are equal. So, $\overline{MA}\cong\overline{HT}$.

Answer:

$\overline{MA}\cong\overline{HT}$ is proved.