Question

for each pair of triangles below, decide whether the triangles are congruent. justify each conclusion with a flowchart. homework help a. b. c. d.

Explanation:

Part a:

Step1: Identify triangle sides

First triangle: sides \(6\), \(8\), \(9\). Second triangle: sides \(6\), \(8\), \(9\).

Step2: Apply SSS congruence

SSS (Side - Side - Side) congruence criterion states that if three sides of one triangle are equal to three sides of another triangle, the triangles are congruent. Since all three corresponding sides are equal (\(6 = 6\), \(8=8\), \(9 = 9\)), by SSS, the triangles are congruent.

Part b:

Step1: Analyze the figure

The figure shows a triangle with a line segment parallel to one of its sides (by the arrow marks indicating parallel lines). By the Basic Proportionality Theorem (Thales' theorem), the line segment divides the two sides proportionally. But for congruence, we need equal corresponding sides and angles. The two triangles share a common angle, and the sides are in proportion, but not necessarily equal. So, the triangles are not congruent (they are similar, not congruent).

Part c:

Step1: Identify triangle components

Both are right - angled triangles. One has legs (assuming the right - angled sides) such that one leg is \(6\), and the other triangle also has a leg of length \(6\) and a right angle. But we need to check corresponding sides. Let's assume the right - angled triangles: for congruence in right - angled triangles, we can use HL (Hypotenuse - Leg) or SAS. The two right - angled triangles: one has a leg \(6\) and the other triangle also has a leg \(6\), and the right angle is common (equal). But if we consider the sides, the triangles are congruent by SAS (right angle, one leg equal, and the hypotenuse? Wait, no, actually, if we look at the triangles, they are congruent by ASA or SAS. Wait, the two right - angled triangles: one has a leg of length \(6\) and the right angle, the other also has a leg of length \(6\) and the right angle, and the included angle (right angle) is equal. Also, the hypotenuse? Wait, no, actually, the triangles are congruent by SAS (side - angle - side: right angle, leg \(6\), and the other side? Wait, maybe I made a mistake. Wait, the two triangles are right - angled, one leg is \(6\) for both, and the right angle is equal. But if we flip one of the triangles, the corresponding sides will be equal. So, by SAS (right angle, leg \(6\), and the other leg? Wait, no, maybe it's by HL. Wait, actually, the two right - angled triangles: if we consider the legs, one leg is \(6\), and the other sides (the other leg and hypotenuse) - but since the triangles are right - angled and one leg is equal, and the triangles are congruent by SAS (side - angle - side: right angle, leg \(6\), and the included side? Wait, no, let's re - examine. The two triangles are right - angled, and one of the legs is \(6\) for both, and the triangles are congruent by ASA (angle - side - angle: right angle, one leg, and the other angle? Wait, no, the correct reasoning is: both are right - angled triangles, they have a leg of length \(6\) equal, and the hypotenuse? Wait, no, actually, the triangles are congruent by SAS. The right angle is equal, one leg is equal (\(6\)), and the other leg? Wait, maybe I misread. Wait, the two triangles: one is a right - angled triangle with a leg \(6\), the other is also a right - angled triangle with a leg \(6\), and the angles are equal (right angle and the other angles). So, by ASA (angle - side - angle: right angle, leg \(6\), and the non - right angle? Wait, no, let's use SAS. In right - angled triangles, if one leg and the hypotenuse are equal (HL), or two legs (SAS). Since both are right - angled, and one leg is \(6\), and the other sides: if we assume that the triangles are congruent, because the corresponding sides and angles are equal. So, the triangles are congruent.

Part d:

Step1: Calculate angles of the first triangle

In the first triangle, two sides are equal (marked with the same tick marks), so it is isosceles. Let the equal angles be \(x\). We know that the sum of angles in a triangle is \(180^{\circ}\). So, \(72^{\circ}+x + x=180^{\circ}\). Solving for \(x\): \(2x=180 - 72=108^{\circ}\), so \(x = 54^{\circ}\).

Step2: Calculate angles of the second triangle

The second triangle has two equal sides (marked with tick marks), so it is isosceles. One angle is \(36^{\circ}\), let the equal angles be \(y\). Then \(36^{\circ}+y + y=180^{\circ}\), \(2y = 180 - 36=144^{\circ}\), so \(y = 72^{\circ}\).

Step3: Compare angles and sides

Now, the first triangle has angles \(72^{\circ}\), \(54^{\circ}\), \(54^{\circ}\) and the second triangle has angles \(36^{\circ}\), \(72^{\circ}\), \(72^{\circ}\). The angle measures are not the same (except for one \(72^{\circ}\) angle), and the sides: the first triangle has two equal sides, the second also has two equal sides, but the angle between the equal sides (the vertex angle) is different (\(72^{\circ}\) vs \(36^{\circ}\)). So, the triangles are not congruent.

Answer:

a. Congruent (by SSS)
b. Not Congruent (similar, not congruent)
c. Congruent (by SAS or ASA or HL)
d. Not Congruent