1. **Problem 2(a): Write a congruency statement for the two triangles sharing line segment.** Given two triangles sharing a side, congruency can be stated by matching corresponding vertices in order. 2. **Problem 2(b): Write a congruency statement for the parallelogram-like quadrilateral with diagonal.** Identify corresponding vertices and state congruency accordingly. 3. **Problem 3(a): Find unknown sides $x$ and $y$ in similar triangles with sides 11 cm, 13 cm, 16 cm, $x$, $y$, and 9 cm.** Step 1: State the similarity ratio between corresponding sides. Step 2: Use the ratio to find $x$ and $y$. 4. **Problem 3(b): Find unknown side $p$ in similar triangles with sides 10 cm, 8 cm, 9 cm, and $p$.** Step 1: Identify corresponding sides. Step 2: Set up proportion and solve for $p$. 5. **Problem 4(a): Find unknown sides $f$ and $d$ in similar triangles with sides 4 cm, 6 cm, 7 cm, $f$, 12 cm, and $d$.** Step 1: Write similarity ratio. Step 2: Solve for $f$ and $d$. 6. **Problem 4(b): Find unknown sides $s$ and $r$ in similar triangles with sides 10 cm, 8 cm, $s$, 18 cm, 24 cm, and $r$.** Step 1: Write similarity ratio. Step 2: Solve for $s$ and $r$. 7. **Problem 4(c): Find unknown sides $w$ and $b$ in similar triangles with sides 4 cm, 5 cm, 7 cm, 9 cm, 6 cm, $w$, and $b$.** Step 1: Write similarity ratio. Step 2: Solve for $w$ and $b$. 8. **Problem 5(a): Find $x$ in the figure with points P, Q, R, S, T and given side lengths.** Step 1: Use properties of intersecting lines and triangles. Step 2: Set up proportion and solve for $x$. 9. **Problem 5(b): Find $x$ in triangle ABC with segment DE parallel to BC.** Step 1: Use the Triangle Proportionality Theorem. Step 2: Set up proportion and solve for $x$. --- ### Detailed solution for Problem 2(a): **Problem:** Write a congruency statement for the two triangles sharing a line segment with vertices A, B, C, D, E. **Step 1:** Identify the two triangles. Suppose they are $\triangle ABC$ and $\triangle DEC$ sharing side $\overline{EC}$. **Step 2:** Check corresponding vertices and sides. **Step 3:** Write the congruency statement matching corresponding vertices in order. **Answer:** $\triangle ABC \cong \triangle DEC$ --- ### Detailed solution for Problem 3(a): **Problem:** Triangles are similar. Find $x$ and $y$. Given sides: - Larger triangle: 11 cm, 13 cm, 16 cm - Smaller triangle: $x$, $y$, 9 cm **Step 1:** Identify corresponding sides. Assume 16 cm corresponds to 9 cm. **Step 2:** Write similarity ratio: $$ \frac{16}{9} = \frac{11}{x} = \frac{13}{y} $$ **Step 3:** Solve for $x$: $$ x = \frac{11 \times 9}{16} = \frac{99}{16} = 6.2 $$ **Step 4:** Solve for $y$: $$ y = \frac{13 \times 9}{16} = \frac{117}{16} = 7.3 $$ **Answer:** $x = 6.2$ cm, $y = 7.3$ cm --- ### Detailed solution for Problem 4(a): **Problem:** Find $f$ and $d$ in similar triangles with sides 4 cm, 6 cm, 7 cm and $f$, 12 cm, $d$. **Step 1:** Write similarity ratio: $$ \frac{f}{4} = \frac{12}{6} = \frac{d}{7} $$ **Step 2:** Simplify ratio: $$ \frac{12}{6} = 2 $$ **Step 3:** Solve for $f$: $$ f = 4 \times 2 = 8 $$ **Step 4:** Solve for $d$: $$ d = 7 \times 2 = 14 $$ **Answer:** $f = 8$ cm, $d = 14$ cm --- ### Detailed solution for Problem 5(b): **Problem:** Find $x$ in triangle ABC with segment DE parallel to BC, $AD = 6$ cm, $DB = x$, $DE = 4$ cm, and $BC = 10$ cm. **Step 1:** By Triangle Proportionality Theorem: $$ \frac{AD}{DB} = \frac{DE}{BC} $$ **Step 2:** Substitute known values: $$ \frac{6}{x} = \frac{4}{10} $$ **Step 3:** Cross multiply: $$ 6 \times 10 = 4 \times x $$ $$ 60 = 4x $$ **Step 4:** Solve for $x$: $$ x = \frac{60}{4} = 15 $$ **Answer:** $x = 15$ cm