**Given:** $ M $ is the midpoint of $ \overline{HS} $ and $ \overline{GT} $.**Prove:** $\triangle GMH \cong \triangle TMS$The diagram shows two triangles, $ \triangle GMH $ and $ \triangle TMS $, intersecting at a common point $ M $, with lines $\overline{HS}$ and $\overline{GT}$.**Proof:**1. **$M$ is the midpoint of $\overline{HS}$ and $\overline{GT}$.** *Reason:* Given2. **$GM \cong TM$** *Reason:* As $M$ is the midpoint, $ \overline{GM} = \overline{TM} $.3. **$HM \cong SM$** *Reason:* As $M$ is the midpoint, $ \overline{HM} = \overline{SM} $.4. **$\angle GMH \cong \angle TMS$** *Reason:* Vertically opposite angles are equal.5. **$\triangle GMH \cong \triangle TMS$** *Reason:* By SAS (Side-Angle-Side) congruence criterion.

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Description: **Given:** \( M $ is the midpoint of $ \overline{HS} $ and $ \overline{GT} $.**Prove:** $\triangle GMH \cong \triangle TMS$The diagram shows two triangles, $ \triangle GMH $ and $ \triangle TMS $, intersecting at a common point $ M $, w

It is given that M is the midpoint of both the segments HS and GT. Therefore, GM≅TM since M is the…

Answered: Given: M is the midpoint of HS and GT.…

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Given: M is the midpoint of HS and GT. Prove: ΔGΜΗ ΔΤM H 1) Mis the midpoint of HS and GT. 1) 2) GM = TM 2) 3) HM = SM 3) 4) ZGMH = ZTMS 4) 5) AGMH = ATMS 5)

Elementary Geometry For College Students, 7e

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ISBN:9781337614085

Author:Alexander, Daniel C.; Koeberlein, Geralyn M.

Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.

ChapterP: Preliminary Concepts

SectionP.CT: Test

Problem 1CT

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Question

**Given:** $ M $ is the midpoint of $ \overline{HS} $ and $ \overline{GT} $.

**Prove:** $\triangle GMH \cong \triangle TMS$

The diagram shows two triangles, $ \triangle GMH $ and $ \triangle TMS $, intersecting at a common point $ M $, with lines $\overline{HS}$ and $\overline{GT}$.

**Proof:**

1. **$M$ is the midpoint of $\overline{HS}$ and $\overline{GT}$.**
*Reason:* Given

2. **$GM \cong TM$**
*Reason:* As $M$ is the midpoint, $ \overline{GM} = \overline{TM} $.

3. **$HM \cong SM$**
*Reason:* As $M$ is the midpoint, $ \overline{HM} = \overline{SM} $.

4. **$\angle GMH \cong \angle TMS$**
*Reason:* Vertically opposite angles are equal.

5. **$\triangle GMH \cong \triangle TMS$**
*Reason:* By SAS (Side-Angle-Side) congruence criterion.

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