Question

example 1

1. camping in the diagram of the pup tent, the

support pole is perpendicular to the ground. the
base of the support pole is located at the midpoint
of the segment connecting the bottom of the sides
of the tent. write a two - column proof to show that the
triangles formed by the support pole are congruent.
given: $overline{xz} perp overline{wy}$; $z$ is the midpoint of $overline{wy}$.
prove: $\triangle wxz cong \triangle yxz$

1. towers the cell phone tower has parallel poles and diagonal support

beams that form two triangles. write a two - column proof to show that the
triangles are congruent.
given: $angle h$ and $angle k$ are right angles
$overline{gh} parallel overline{kj}$
prove: $\triangle gkj cong \triangle jhg$

Explanation:

Problem 1: CAMPING (Two - column proof for $\triangle WXZ\cong\triangle YXZ$)

Step 1: Identify the midpoint property

Since $Z$ is the midpoint of $\overline{WY}$, by the definition of a midpoint, we have $\overline{WZ}\cong\overline{YZ}$.

Step 2: Identify the right angles

Given that $\overline{XZ}\perp\overline{WY}$, by the definition of perpendicular lines, $\angle WXZ$ and $\angle YXZ$ are right angles. So, $\angle WXZ\cong\angle YXZ$ (all right angles are congruent).

Step 3: Identify the common side

The segment $\overline{XZ}$ is common to both $\triangle WXZ$ and $\triangle YXZ$, so $\overline{XZ}\cong\overline{XZ}$ (reflexive property of congruence).

Step 4: Apply the SAS congruence criterion

We have $\overline{WZ}\cong\overline{YZ}$, $\angle WXZ\cong\angle YXZ$, and $\overline{XZ}\cong\overline{XZ}$. By the Side - Angle - Side (SAS) congruence postulate, $\triangle WXZ\cong\triangle YXZ$.

Statements	Reasons
2. $\overline{WZ}\cong\overline{YZ}$	Definition of midpoint
3. $\overline{XZ}\perp\overline{WY}$	Given
4. $\angle WXZ$ and $\angle YXZ$ are right angles	Definition of perpendicular lines
5. $\angle WXZ\cong\angle YXZ$	All right angles are congruent
6. $\overline{XZ}\cong\overline{XZ}$	Reflexive property of congruence
7. $\triangle WXZ\cong\triangle YXZ$	SAS (Steps 2, 5, 6)

Problem 2: TOWERS (Two - column proof for $\triangle GKJ\cong\triangle JHG$)

(Assuming the diagram has $\overline{GH}\parallel\overline{KJ}$, $\angle H$ and $\angle K$ are right angles, and we are proving $\triangle GKJ\cong\triangle JHG$)

Step 1: Identify the right angles

Given that $\angle H$ and $\angle K$ are right angles, so $\angle H\cong\angle K$ (all right angles are congruent).

Step 2: Identify the alternate interior angles

Since $\overline{GH}\parallel\overline{KJ}$, by the alternate interior angles theorem, $\angle G\cong\angle J$ (alternate interior angles are congruent when lines are parallel).

Step 3: Identify the common side

The segment $\overline{GJ}$ (or the appropriate common side, assuming $\overline{GJ}$ is common or there is a common side from the diagram context) or if we assume the triangles share a side like $\overline{GJ}\cong\overline{JG}$ (reflexive property). Also, we can consider the sides formed by the parallel lines and the right angles. If we assume the length of $\overline{GH}$ and $\overline{KJ}$ are related by the parallel lines and the right angles, and we have a common side. Let's assume the common side is $\overline{GJ}\cong\overline{JG}$ (reflexive property).

Step 4: Apply the AAS congruence criterion

We have $\angle H\cong\angle K$, $\angle G\cong\angle J$, and $\overline{GJ}\cong\overline{JG}$. By the Angle - Angle - Side (AAS) congruence postulate, $\triangle GKJ\cong\triangle JHG$.

Statements	Reasons
2. $\angle H\cong\angle K$	All right angles are congruent
3. $\overline{GH}\parallel\overline{KJ}$	Given
4. $\angle G\cong\angle J$	Alternate Interior Angles Theorem
5. $\overline{GJ}\cong\overline{JG}$	Reflexive property of congruence
6. $\triangle GKJ\cong\triangle JHG$	AAS (Steps 2, 4, 5)

Answer:

For Problem 1:

The two - column proof as shown above proves that $\triangle WXZ\cong\triangle YXZ$ by the SAS congruence criterion.

For Problem 2:

The two - column proof as shown above (assuming the diagram - related properties) proves that $\triangle GKJ\cong\triangle JHG$ by the AAS congruence criterion.