11). solve for x:
12). list the (5) triangle congruence (abbreviations).
13). what does cpctc mean? write out what the acronym means - spelling counts!
14). in the figure below, w is the midpoint of $ \overline { t v } $ and $ \overline { u w } \perp \overline { t v } $. mark the triangle with congruen marks and state which congruence theorem proves that $ \delta u v w \cong \delta u t w $?
15). write a congruence statement and that justifies the triangles congruent by sas.

Question

Accepted Answer

### 11) Solve for $ x $
# Explanation:
## Step1: Recognize the triangle type
The triangle is equilateral (all sides equal). In an equilateral triangle, all angles are $ 60^{\circ} $.
## Step2: Set up the equation
We have $ 7x + 4=60 $.
## Step3: Solve the equation
Subtract $ 4 $ from both sides: $ 7x=60 - 4=56 $.
Divide both sides by $ 7 $: $ x=\frac{56}{7}=8 $.

# Answer:
$ x = 8 $

### 12) List the (5) triangle congruence (abbreviations)
# Brief Explanations:
Triangle congruence abbreviations are based on geometric theorems.
- **SSS (Side - Side - Side)**: If three sides of one triangle are equal to three sides of another triangle, the triangles are congruent.
- **SAS (Side - Angle - Side)**: If two sides and the included angle of one triangle are equal to two sides and the included angle of another triangle, the triangles are congruent.
- **ASA (Angle - Side - Angle)**: If two angles and the included side of one triangle are equal to two angles and the included side of another triangle, the triangles are congruent.
- **AAS (Angle - Angle - Side)**: If two angles and a non - included side of one triangle are equal to two angles and the corresponding non - included side of another triangle, the triangles are congruent.
- **HL (Hypotenuse - Leg) (for right - angled triangles)**: If the hypotenuse and a leg of one right - angled triangle are equal to the hypotenuse and a leg of another right - angled triangle, the triangles are congruent.

# Answer:
SSS, SAS, ASA, AAS, HL

### 13) What does CPCTC mean?
# Brief Explanations:
CPCTC is an acronym used in geometry. It is based on the property that when two triangles are congruent, their corresponding parts are also congruent.

# Answer:
Corresponding Parts of Congruent Triangles are Congruent

### 14) Mark the triangle with congruence marks and state which congruence theorem proves that $ \triangle UVW\cong\triangle UTW $
# Brief Explanations:
- Since $ W $ is the midpoint of $ \overline{TV} $, $ TW = VW $.
- $ \overline{UW}\perp\overline{TV} $, so $ \angle UWT=\angle UWV = 90^{\circ} $.
- $ \overline{UW} $ is common to both $ \triangle UTW $ and $ \triangle UVW $.
- By the SAS (Side - Angle - Side) congruence theorem ( $ TW = VW $, $ \angle UWT=\angle UWV $, $ UW = UW $), the triangles are congruent.

# Answer:
Mark $ TW\cong VW $ (mid - point), $ \angle UWT\cong\angle UWV $ (right angles), $ UW\cong UW $ (common side). The congruence theorem is SAS.

### 15) Write a congruence statement and that justifies the triangles congruent by SAS
# Brief Explanations:
- In $ \triangle PRS $ and $ \triangle PQS $:
    - $ PR = PQ $ (given as marked sides).
    - $ \angle RPS=\angle QPS $ (given as marked angles).
    - $ PS = PS $ (common side).

# Answer:
Congruence statement: $ \triangle PRS\cong\triangle PQS $. Justification: SAS ( $ PR = PQ $, $ \angle RPS=\angle QPS $, $ PS = PS $)

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QUESTION IMAGE

Question

Explanation:

11) Solve for \( x \)

Step1: Recognize the triangle type

Step2: Set up the equation

Step3: Solve the equation

Answer:

12) List the (5) triangle congruence (abbreviations)