3. a. is △xyw ~ △xwz? explain why or why not.
b. is △wyz ~ △xwz? explain why or why not.
c. is △xyw ~ △wyz? explain why or why not.
4. in the figure at the right, ∠c≅∠f.
a. which triangle is similar to △bcd?
b. which triangle is similar to △ace?
5. can you use the information in the figure to show that the triangles are similar? explain. if so, find the value of x.
6. can you use the information in the figure to show that the triangles are similar? explain. if so, find the value of x.
154 lesson 7 describe angle relationships in triangles

Question

Accepted Answer

### 3a.
# Explanation:
## Step1: Identify equal angles
In $\triangle XYW$ and $\triangle XWZ$, $\angle XYW=\angle XWZ = 90^{\circ}$, and $\angle X$ is common to both triangles.
## Step2: Apply AA - similarity criterion
Since two angles of $\triangle XYW$ are equal to two angles of $\triangle XWZ$ (angle - angle similarity), $\triangle XYW\sim\triangle XWZ$.
# Answer:
Yes, because $\angle XYW=\angle XWZ = 90^{\circ}$ and $\angle X$ is common.

### 3b.
# Explanation:
## Step1: Analyze angle - angle relationships
In $\triangle WYZ$ and $\triangle XWZ$, $\angle WYZ = 90^{\circ}$ and $\angle XWZ=90^{\circ}$, but $\angle Z$ is common to $\triangle WYZ$ and $\triangle XWZ$, and $\angle WZY
eq\angle XZW$ in general. Also,  $\angle YWZ
eq\angle XWZ$ (except in some special cases which are not given here).
## Step2: Determine similarity
There are not two pairs of equal angles, so $\triangle WYZ$ is not similar to $\triangle XWZ$.
# Answer:
No, as there are not two pairs of equal angles.

### 3c.
# Explanation:
## Step1: Check angle - angle condition
In $\triangle XYW$ and $\triangle WYZ$, $\angle XYW=\angle WYZ = 90^{\circ}$. But we do not have enough information to show that another pair of angles are equal. For example, we do not know the measures of non - right angles $\angle XWY$ and $\angle WZY$ and there is no information to suggest they are equal.
## Step2: Conclude similarity
So, $\triangle XYW$ is not similar to $\triangle WYZ$.
# Answer:
No, as there is not enough information to show two pairs of equal angles.

### 4a.
# Explanation:
## Step1: Use the AA - similarity criterion
Since $\angle C\cong\angle F$ and $\angle BDC\cong\angle EDF$ (vertically opposite angles), by the AA - similarity criterion, $\triangle BCD\sim\triangle EFD$.
# Answer:
$\triangle EFD$

### 4b.
# Explanation:
## Step1: Apply AA - similarity
Since $\angle C\cong\angle F$ and $\angle A$ is common to $\triangle ACE$ and $\triangle AFD$, by the AA - similarity criterion, $\triangle ACE\sim\triangle AFD$.
# Answer:
$\triangle AFD$

### 5
# Explanation:
## Step1: Check for similar triangles
The two right - angled triangles have a pair of vertical angles which are equal. Let the vertical angles be $\alpha$. So, by the AA - similarity criterion, the two triangles are similar.
## Step2: Set up proportion
The ratio of corresponding sides of similar triangles is equal. We have $\frac{x}{9}=\frac{20}{15}$.
## Step3: Solve for $x$
Cross - multiply: $15x=20\times9$, then $15x = 180$, and $x=\frac{180}{15}=12$.
# Answer:
Yes, the triangles are similar. $x = 12$

### 6
# Explanation:
## Step1: Check for similar triangles
We can use the AA - similarity criterion. The two triangles have a pair of equal angles (the non - marked angles which are equal because the other non - right angles are formed by parallel lines and transversals or vertical - angle - like relationships). Let the equal non - right angles be $\theta$.
## Step2: Set up proportion
The ratio of corresponding sides of similar triangles gives $\frac{4}{7}=\frac{5}{x}$.
## Step3: Solve for $x$
Cross - multiply: $4x=7\times5$, so $4x = 35$, and $x=\frac{35}{4}=8.75$.
# Answer:
Yes, the triangles are similar. $x = 8.75$

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QUESTION IMAGE

Question

Explanation:

3a.

Step1: Identify equal angles

Step2: Apply AA - similarity criterion

Step1: Analyze angle - angle relationships

Step2: Determine similarity

Step1: Check angle - angle condition

Step2: Conclude similarity

Answer:

3b.