II. Given $$ riangle A R T \cong riangle H O P $$, if $$ m \angle R=54^{\circ} $$ and $$ m \angle T=73^{\circ}, H O=15 \mathrm{~cm} $$. $$ \mathrm{AT}=10 \mathrm{~cm} $$. 1. Ilustrate the two triangles. 2. What angles of $$ riangle H O P $$ have a measure of $$ 53^{\circ} $$ and a measure of $$ 54^{\circ} $$ a 3. Identify the sides with measures equal to HO and AT . 4. If $$ R T=20 \mathrm{~cm} $$. find the value of $$ imes $$ if $$ O P=11 imes-13 $$ 5. Which is the shortest side of $$ riangle A R T F $$

Question

II. Given $$ 	riangle A R T \cong 	riangle H O P $$, if $$ m \angle R=54^{\circ} $$ and $$ m \angle T=73^{\circ}, H O=15 \mathrm{~cm} $$. $$ \mathrm{AT}=10 \mathrm{~cm} $$. 1. Ilustrate the two triangles. 2. What angles of $$ 	riangle H O P $$ have a measure of $$ 53^{\circ} $$ and a measure of $$ 54^{\circ} $$ a 3. Identify the sides with measures equal to HO and AT . 4. If $$ R T=20 \mathrm{~cm} $$. find the value of $$ 	imes $$ if $$ O P=11 	imes-13 $$ 5. Which is the shortest side of $$ 	riangle A R T F $$

UpStudy AI · Accepted Answer

Certainly! Let's address each part of the problem step by step.

---

### **Given:**
- **Triangles:** $$ \triangle A R T $$ is congruent to $$ \triangle H O P $$ (i.e., $$ \triangle A R T \cong \triangle H O P $$)
- **Angles in $$ \triangle A R T $$:**
  - $$ m \angle R = 54^\circ $$
  - $$ m \angle T = 73^\circ $$
  - Therefore, $$ m \angle A = 180^\circ - 54^\circ - 73^\circ = 53^\circ $$
- **Sides:**
  - $$ HO = 15 $$ cm
  - $$ AT = 10 $$ cm
  - $$ RT = 20 $$ cm (given in part 4)

---

### **1. Illustrate the Two Triangles**

Since I can't provide a visual diagram, I'll describe how the two congruent triangles $$ \triangle A R T $$ and $$ \triangle H O P $$ correspond to each other.

**Triangle $$ \triangle A R T $$:**
- **Vertices:** A, R, T
- **Angles:**
  - $$ \angle A = 53^\circ $$
  - $$ \angle R = 54^\circ $$
  - $$ \angle T = 73^\circ $$
- **Sides:**
  - $$ AT = 10 $$ cm
  - $$ RT = 20 $$ cm
  - $$ AR = $$ (to be determined)

**Triangle $$ \triangle H O P $$:**
- **Vertices:** H, O, P
- **Angles:**
  - $$ \angle H = 53^\circ $$ (corresponding to $$ \angle A $$)
  - $$ \angle O = 54^\circ $$ (corresponding to $$ \angle R $$)
  - $$ \angle P = 73^\circ $$ (corresponding to $$ \angle T $$)
- **Sides:**
  - $$ HO = 15 $$ cm (corresponding to $$ AR $$)
  - $$ OP = 20 $$ cm (corresponding to $$ RT $$)
  - $$ HP = 10 $$ cm (corresponding to $$ AT $$)

**Correspondence Summary:**

| $$ \triangle A R T $$ | $$ \triangle H O P $$ |
|----------------------|----------------------|
| Vertex A             | Vertex H             |
| Vertex R             | Vertex O             |
| Vertex T             | Vertex P             |
| Side AT = 10 cm      | Side HP = 10 cm      |
| Side RT = 20 cm      | Side OP = 20 cm      |
| Side AR = 15 cm      | Side HO = 15 cm      |

---

### **2. Determine the Measures of the Angles in $$ \triangle H O P $$**

Since $$ \triangle A R T \cong \triangle H O P $$, corresponding angles are equal.

- $$ \angle H $$ corresponds to $$ \angle A $$: $$ m \angle H = 53^\circ $$
- $$ \angle O $$ corresponds to $$ \angle R $$: $$ m \angle O = 54^\circ $$
- $$ \angle P $$ corresponds to $$ \angle T $$: $$ m \angle P = 73^\circ $$

**Answer:**
- $$ \angle H = 53^\circ $$
- $$ \angle O = 54^\circ $$
- $$ \angle P = 73^\circ $$

---

### **3. Identify the Sides with Measures Equal to $$ HO $$ and $$ AT $$**

Given the congruence $$ \triangle A R T \cong \triangle H O P $$, corresponding sides are equal.

- **Side $$ HO = 15 $$ cm** corresponds to **Side $$ AR = 15 $$ cm**.
- **Side $$ AT = 10 $$ cm** corresponds to **Side $$ HP = 10 $$ cm**.

**Summary:**
- $$ HO $$ corresponds to $$ AR $$
- $$ AT $$ corresponds to $$ HP $$

---

### **4. Find the Value of $$ x $$ Given $$ OP = 11x - 13 $$ cm**

Given:
- $$ RT = 20 $$ cm in $$ \triangle A R T $$
- Since $$ \triangle A R T \cong \triangle H O P $$, the corresponding side $$ OP $$ in $$ \triangle H O P $$ is also $$ 20 $$ cm.

Set up the equation:
$$ OP = 11x - 13 $$
$$ 20 = 11x - 13 $$

**Solve for $$ x $$:**
$$
\begin{align*}
20 &= 11x - 13 \
20 + 13 &= 11x \
33 &= 11x \
x &= \frac{33}{11} \
x &= 3
\end{align*}
$$

**Answer:** $$ x = 3 $$

---

### **5. Determine the Shortest Side of $$ \triangle A R T F $$**

There seems to be a typo in the question since $$ \triangle A R T F $$ suggests a quadrilateral rather than a triangle. However, assuming it's meant to be $$ \triangle A R T $$, we'll determine the shortest side of $$ \triangle A R T $$.

**Given:**
- $$ AT = 10 $$ cm
- $$ RT = 20 $$ cm
- $$ AR = 15 $$ cm (from part 3)

**Side Lengths:**
- $$ AT = 10 $$ cm
- $$ AR = 15 $$ cm
- $$ RT = 20 $$ cm

**Shortest Side:** $$ AT = 10 $$ cm

**Answer:** The shortest side of $$ \triangle A R T $$ is $$ AT $$, which measures **10 cm**.

---

If the mention of $$ \triangle A R T F $$ was intentional and refers to a different figure or if "F" represents another point forming a different triangle, please provide additional information or clarify the context so I can assist you accurately.
1. **Illustrate the Two Triangles:** - Draw $$ \triangle A R T $$ with angles $$ 53^\circ $$, $$ 54^\circ $$, and $$ 73^\circ $$, and sides $$ AT = 10 $$ cm, $$ RT = 20 $$ cm, and $$ AR = 15 $$ cm. - Draw $$ \triangle H O P $$ with corresponding angles and sides: $$ \angle H = 53^\circ $$, $$ \angle O = 54^\circ $$, $$ \angle P = 73^\circ $$, $$ HO = 15 $$ cm, $$ OP = 20 $$ cm, and $$ HP = 10 $$ cm. 2. **Angles in $$ \triangle H O P $$:** - $$ \angle H = 53^\circ $$ - $$ \angle O = 54^\circ $$ - $$ \angle P = 73^\circ $$ 3. **Identify Corresponding Sides:** - $$ HO = 15 $$ cm corresponds to $$ AR = 15 $$ cm - $$ AT = 10 $$ cm corresponds to $$ HP = 10 $$ cm - $$ OP = 20 $$ cm corresponds to $$ RT = 20 $$ cm 4. **Find $$ x $$:** - Given $$ OP = 11x - 13 $$ cm and $$ OP = 20 $$ cm - Solve: $$ 11x - 13 = 20 $$ → $$ x = 3 $$ 5. **Shortest Side of $$ \triangle A R T $$:** - The shortest side is $$ AT = 10 $$ cm

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