Question

1. if two triangles have two pairs of corresponding angles that are equal, which criterion can be used to prove they are similar?

a. angle-angle-angle
b. side-side-side
c. angle-angle
d. angle-side-angle

1. what does the midline theorem state about the segment joining the midpoints of two sides of a triangle?

a. it is perpendicular to the third side.
b. it is parallel to the third side and half its length.
c. it is equal in length to the third side.
d. it bisects the third side.

1. a triangular garden has a fence parallel to its base that divides one side into 7 m and 5 m segments and the other side into 14 m and y segments, respectively. what is y?

a. 7 m
b. 5 m
c. 10 m
d. 8 m

Explanation:

Question 7

To determine the similarity criterion for two triangles with two pairs of equal corresponding angles:

• Recall triangle similarity criteria:
• AAA (Angle - Angle - Angle) requires three pairs of equal angles. But if two angles are equal, the third must also be equal (since the sum of angles in a triangle is \(180^{\circ}\)), but the criterion using two angles is called AA (Angle - Angle).
• SSS (Side - Side - Side) is about proportional sides, not angles.
• SAS (Side - Angle - Side) is about a pair of equal angles and proportional sides around the angle.
• The AA (Angle - Angle) criterion states that if two angles of one triangle are equal to two angles of another triangle, the triangles are similar. Since two pairs of equal angles are given, the AA criterion applies.

Recall the Midline Theorem (also known as the Midsegment Theorem) for a triangle:

• The segment joining the midpoints of two sides of a triangle is parallel to the third side and half as long as the third side.
• Option a is incorrect as the midline is parallel, not perpendicular.
• Option c is incorrect as it is half the length, not equal.
• Option d is incorrect as bisecting the third side is not the main statement of the midline theorem (it is parallel and half the length).

Step 1: Identify the theorem

Since the fence is parallel to the base of the triangular garden, we can use the Basic Proportionality Theorem (Thales' theorem), which states that if a line is drawn parallel to one side of a triangle, intersecting the other two sides, then it divides those sides proportionally.
So, if one side is divided into segments of length \(7\space m\) and \(5\space m\), and the other side is divided into segments of length \(14\space m\) and \(y\), we have the proportion \(\frac{7}{5}=\frac{14}{y}\) (or \(\frac{7}{14}=\frac{5}{y}\), depending on the correspondence of the segments. Let's use the correct proportion: if the line parallel to the base divides the first side into \(a = 7\), \(b=5\) and the second side into \(c = 14\), \(d = y\), then \(\frac{a}{b}=\frac{c}{d}\) (assuming the segments are in the same order). Wait, actually, the correct proportion from Thales' theorem is \(\frac{\text{segment 1 on side 1}}{\text{segment 2 on side 1}}=\frac{\text{segment 1 on side 2}}{\text{segment 2 on side 2}}\). So \(\frac{7}{5}=\frac{14}{y}\) is incorrect. The correct proportion is \(\frac{7}{14}=\frac{5}{y}\) (because the ratio of the segments on one side should equal the ratio of the segments on the other side). Let's solve \(\frac{7}{14}=\frac{5}{y}\).

Step 2: Solve for \(y\)

Cross - multiply: \(7\times y=14\times5\)
Simplify: \(7y = 70\)
Divide both sides by 7: \(y=\frac{70}{7}=10\)
Wait, no. Wait, the correct proportion is \(\frac{7}{5}=\frac{14}{y}\)? Wait, let's think again. If the fence is parallel to the base, then the two smaller triangles (formed by the fence) and the original triangle are similar. So the ratio of the corresponding sides should be equal. Let the two segments on the first side be \(x_1 = 7\) and \(x_2=5\), so the total length of the first side is \(7 + 5=12\). The two segments on the second side are \(y_1 = 14\) and \(y_2=y\), so the total length of the second side is \(14 + y\). Since the triangles are similar, \(\frac{7}{14}=\frac{5}{y}\) (because the ratio of the upper segment to the lower segment on one side is equal to the ratio of the upper segment to the lower segment on the other side). Solving \(\frac{7}{14}=\frac{5}{y}\):
Cross - multiply: \(7y=14\times5\)
\(7y = 70\)
\(y = 10\). Wait, but let's check the other way. If we consider the ratio of the segments as \(\frac{7}{5}=\frac{14}{y}\), then \(7y=70\), \(y = 10\) as well. So the value of \(y\) is \(10\space m\).

Answer:

c. Angle - Angle

Question 8