Question

similarity - same shape, different size
3 types
aa (congruent angles)
sas (2 ratios, one angle)
sss (3 side ratios)

1. δvut - δefg
2. δuts - δcde

Explanation:

Problem 3: $\triangle VUT \sim \triangle EFG$

Step 1: Identify corresponding sides

For similar triangles $\triangle VUT$ and $\triangle EFG$, we need to find the ratios of corresponding sides. Let's list the side lengths:

• $\triangle VUT$: $VU = 35$, $UT = 49$, $VT = 42$
• $\triangle EFG$: $EF = 20$, $FG = 28$, $EG = 24$

Step 2: Calculate ratios of corresponding sides

• Ratio of $EF$ to $VU$: $\frac{EF}{VU} = \frac{20}{35} = \frac{4}{7}$
• Ratio of $EG$ to $VT$: $\frac{EG}{VT} = \frac{24}{42} = \frac{4}{7}$
• Ratio of $FG$ to $UT$: $\frac{FG}{UT} = \frac{28}{49} = \frac{4}{7}$

Since all ratios are equal ($\frac{4}{7}$), the triangles are similar by SSS (Side - Side - Side) similarity criterion.

Problem 4: $\triangle UTS \sim \triangle CDE$

Step 1: Identify corresponding angles and sides

We know that $\angle U = 63^{\circ}$ and $\angle C = 63^{\circ}$, so $\angle U=\angle C$. Now, let's list the side lengths:

• $\triangle UTS$: $UT = 72$, $US = 66$
• $\triangle CDE$: $CD = 60$, $CE = 55$

Step 2: Calculate ratios of corresponding sides

• Ratio of $CD$ to $UT$: $\frac{CD}{UT}=\frac{60}{72}=\frac{5}{6}$
• Ratio of $CE$ to $US$: $\frac{CE}{US}=\frac{55}{66}=\frac{5}{6}$

Since we have a pair of equal angles ($\angle U = \angle C$) and the ratios of the including sides are equal ($\frac{5}{6}$), the triangles are similar by SAS (Side - Angle - Side) similarity criterion.

For the SAS Similarity Example (with the 30° angle)

Step 1: Identify sides and angle

We have two triangles. The smaller triangle has sides 5 and 6 with an included 30° angle. The larger triangle has sides 10 and 12 with an included 30° angle.

Step 2: Calculate ratios of corresponding sides

• Ratio of 10 to 5: $\frac{10}{5}=2$
• Ratio of 12 to 6: $\frac{12}{6}=2$

Since the ratio of the two pairs of corresponding sides is equal (2) and the included angle (30°) is equal, the triangles are similar by SAS (Side - Angle - Side) similarity criterion.

For the SSS Similarity Example (with sides 5, 6, 7 and 10, 14,?)

Answer:

Step 1: Identify angles

We have two triangles. Let's assume that in the smaller triangle we can identify two angles (let's say $\angle A$ and $\angle B$) and in the larger triangle, the corresponding angles are equal (because of the AA similarity - if two angles of one triangle are equal to two angles of another triangle, the triangles are similar).

Step 2: Conclude similarity

Since two angles of one triangle are equal to two angles of the other triangle, by the AA (Angle - Angle) similarity criterion, the triangles are similar. This also implies they are similar by AAA (since the sum of angles in a triangle is 180°, if two angles are equal, the third is also equal) and since similar triangles have the same shape, the AA criterion is sufficient for similarity.