Question

e) 1 : 10
a) 1 : 10
c) 5 : 9\t\tb) 1 : 3
e) 4 : 5\t\td) 7 : 10
state if the triangles in each pair are similar. if so, state how you know they are similar.

1. $\triangle jkl \sim \triangle jfg$

a) similar; sas similarity
b) not similar
c) similar; aa similarity
d) similar; sss similarity

1. $\triangle cde \sim \triangle ckj$

a) similar; sss similarity
b) similar; aa similarity
c) not similar
d) similar; sas similarity
18)

Explanation:

Problem 16: $\triangle JKL \sim \triangle JFG$

Step 1: Identify Angles

In the diagram, $\angle KJL$ and $\angle FJG$ are vertical angles (so they are equal). Also, $\angle K$ and $\angle F$ appear to be marked as equal (from the angle symbols), and $\angle L$ and $\angle G$ are also marked as equal. So two pairs of angles are equal.

Step 2: Apply AA Similarity

By the AA (Angle - Angle) Similarity Criterion, if two angles of one triangle are equal to two angles of another triangle, the triangles are similar. Since we have two pairs of equal angles ($\angle K = \angle F$, $\angle L=\angle G$, and vertical angles $\angle KJL=\angle FJG$ confirms the third pair is equal, but AA only needs two), $\triangle JKL \sim \triangle JFG$ by AA similarity.

Problem 17: $\triangle CDE \sim \triangle CKJ$

Step 1: Check Proportions of Sides

First, find the ratios of corresponding sides.

• For side $CK$ and $CD$: Let's see the lengths. $CK = 65$, $CD=169$. The ratio $\frac{CK}{CD}=\frac{65}{169}=\frac{5}{13}$ (dividing numerator and denominator by 13: $65\div13 = 5$, $169\div13 = 13$).
• For side $CJ$ and $CE$: $CJ = 30$, $CE=30 + 78=108$? Wait, no, looking at the diagram, $CJ = 30$, $CE = 78$? Wait, no, the segments: $CJ = 30$, $CE=78$, and $CK = 65$, $CD = 169$. Wait, actually, the vertical side: $CJ = 30$, $CE=78$, so the ratio $\frac{CJ}{CE}=\frac{30}{78}=\frac{5}{13}$ (dividing numerator and denominator by 6: $30\div6 = 5$, $78\div6 = 13$). And the included angle: $\angle KCJ$ and $\angle DCE$ are vertical angles (so equal).

Step 2: Apply SAS Similarity

We have two sides in proportion ($\frac{CK}{CD}=\frac{CJ}{CE}=\frac{5}{13}$) and the included angle (vertical angles) equal. By SAS (Side - Angle - Side) Similarity Criterion, if two sides of one triangle are in proportion to two sides of another triangle and the included angle is equal, the triangles are similar. So $\triangle CDE \sim \triangle CKJ$ by SAS similarity.

Problem 16 Answer:

C) similar; AA similarity

Problem 17 Answer:

D) similar; SAS similarity

Answer:

Step 1: Check Proportions of Sides

First, find the ratios of corresponding sides.

• For side $CK$ and $CD$: Let's see the lengths. $CK = 65$, $CD=169$. The ratio $\frac{CK}{CD}=\frac{65}{169}=\frac{5}{13}$ (dividing numerator and denominator by 13: $65\div13 = 5$, $169\div13 = 13$).
• For side $CJ$ and $CE$: $CJ = 30$, $CE=30 + 78=108$? Wait, no, looking at the diagram, $CJ = 30$, $CE = 78$? Wait, no, the segments: $CJ = 30$, $CE=78$, and $CK = 65$, $CD = 169$. Wait, actually, the vertical side: $CJ = 30$, $CE=78$, so the ratio $\frac{CJ}{CE}=\frac{30}{78}=\frac{5}{13}$ (dividing numerator and denominator by 6: $30\div6 = 5$, $78\div6 = 13$). And the included angle: $\angle KCJ$ and $\angle DCE$ are vertical angles (so equal).

Step 2: Apply SAS Similarity

We have two sides in proportion ($\frac{CK}{CD}=\frac{CJ}{CE}=\frac{5}{13}$) and the included angle (vertical angles) equal. By SAS (Side - Angle - Side) Similarity Criterion, if two sides of one triangle are in proportion to two sides of another triangle and the included angle is equal, the triangles are similar. So $\triangle CDE \sim \triangle CKJ$ by SAS similarity.

Problem 16 Answer:

C) similar; AA similarity

Problem 17 Answer:

D) similar; SAS similarity