Question

sas congruence
name: ____ date: __ period: ____
state if the two triangles are congruent. if they are, state how you know.
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Answer:

Let's take problem 1 as an example to solve (since the problem is about triangle congruence, we'll use the SAS (Side - Angle - Side) congruence criterion. The SAS criterion states that if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent).

Problem 1:

Step 1: Identify the sides and angle

Looking at the first diagram (problem 1), we can see that there is a diagonal in a quadrilateral (which looks like a parallelogram). The two triangles formed by the diagonal:

• The diagonal is a common side, so it is congruent to itself.
• The sides of the parallelogram: In a parallelogram, opposite sides are congruent. So, the two sides adjacent to the diagonal (from each triangle) are congruent.
• The included angle: The angle between the two sides (of each triangle) that are congruent. Since the quadrilateral is a parallelogram, the alternate interior angles (or the angles formed by the diagonal and the sides) are congruent. Also, in a parallelogram, adjacent angles are supplementary, but more importantly, the triangles share the included angle or have congruent included angles due to the properties of the parallelogram. Alternatively, we can see that the two sides and the included angle (formed by the diagonal and the sides of the parallelogram) are congruent for both triangles.

Step 2: Apply SAS congruence

Since we have two sides and the included angle of one triangle congruent to two sides and the included angle of the other triangle, by the SAS (Side - Angle - Side) congruence postulate, the two triangles are congruent.

Problem 2:

Step 1: Identify the sides and angle

In the second diagram (problem 2), we have a triangle divided into two by a line segment (the median or an altitude? Looking at the marks, we can see that there are two sides with the same tick marks (congruent sides) and the included angle (the angle between them) is marked as a right angle (or a congruent angle). The common side (the segment dividing the triangle) is congruent to itself. The two sides from the top vertex to the base and the included angle (the angle between them) are congruent.

Step 2: Apply SAS congruence

By the SAS congruence postulate, since two sides and the included angle of one triangle (formed by the dividing segment) are congruent to two sides and the included angle of the other triangle, the two triangles are congruent.

Problem 3:

Step 1: Identify the sides and angle

In problem 3, we have a triangle divided into two by a segment. We can see that two sides of each triangle (the sides with the same tick marks) and the included angle (the angle between them, which is marked as a congruent angle, maybe a right angle or another congruent angle) are congruent. The common side is also congruent.

Step 2: Apply SAS congruence

Using the SAS congruence postulate, since two sides and the included angle of one triangle are congruent to two sides and the included angle of the other triangle, the two triangles are congruent.

Problem 4:

Step 1: Identify the sides and angle

In problem 4, we have two separate triangles. We can see that there are two sides with tick marks (congruent sides) and the included angle (the angle between them) - one triangle has a right angle and the other also has a right angle (so the included angles are congruent as right angles are congruent, i.e., $90^{\circ}$). Also, the sides with the same tick marks are congruent.

Step 2: Apply SAS congruence

By the SAS congruence postulate, since two sides and the included angle (the right angle) of one triangle are congruent to two sides and the included angle (the right angle) of the other triangle, the two triangles are congruent.

Problem 5:

Step 1: Identify the sides and angle

In problem 5, we have a quadrilateral (a parallelogram - like shape) with a diagonal. The two triangles formed by the diagonal:

• The diagonal is a common side (congruent to itself).
• The sides of the quadrilateral: The opposite sides of the quadrilateral are congruent (since it looks like a parallelogram - like shape with congruent side marks).
• The included angle: The angle between the two sides (of each triangle) that are congruent. The included angle is congruent because of the properties of the quadrilateral (alternate interior angles or angle - side - angle properties, but here we focus on SAS).

Step 2: Apply SAS congruence

By the SAS congruence postulate, the two triangles are congruent.

Problem 6:

Step 1: Identify the sides and angle

In problem 6, we have a quadrilateral divided into two triangles by a diagonal. We can see that there are two sides with the same tick marks (congruent sides) and the included angle (the angle between them) is congruent. The diagonal is a common side (congruent to itself).

Step 2: Apply SAS congruence

By the SAS congruence postulate, the two triangles are congruent.

Problem 7:

Step 1: Identify the sides and angle

In problem 7, we have two separate triangles. Let's check the side and angle marks. The first triangle has sides with tick marks (e.g., one side with two ticks, one with one tick, and one with three ticks). The second triangle has sides with different tick marks. Wait, maybe I mis - see. Wait, no, let's re - examine. If we look at the tick marks, the first triangle has a side with two ticks, a side with one tick, and a side with three ticks. The second triangle has a side with one tick, a side with three ticks, and a side with two ticks? No, maybe the included angle is not congruent. Wait, actually, in problem 7, the sides with the same tick marks: Let's see, the first triangle has a side with two ticks, a side with one tick, and the included angle? The second triangle has a side with one tick, a side with three ticks, and the included angle. Since the sides with the same tick marks are not the two sides with the included angle, the SAS congruence does not apply. Wait, maybe I made a mistake. Wait, no, let's look again. If the first triangle has sides: let's say side $a$ (two ticks), side $b$ (one tick), and included angle $\angle C$. The second triangle has side $b$ (one tick), side $c$ (three ticks), and included angle $\angle D$. Since $a
eq c$ (different tick marks) and the included angles may not be congruent, the two triangles are not congruent by SAS.

Problem 8:

Step 1: Identify the sides and angle

In problem 8, we have a quadrilateral divided into two triangles by a diagonal. We can see that there are two sides with the same tick marks (congruent sides) and the included angle (the angle between them) is congruent. The diagonal is a common side (congruent to itself).

Step 2: Apply SAS congruence

By the SAS congruence postulate, the two triangles are congruent.

Problem 9:

Step 1: Identify the sides and angle

In problem 9, we have a quadrilateral with a diagonal. We can see that there are two sides with the same tick marks (congruent sides) and the included angle (the angle between them) is congruent. The diagonal is a common side (congruent to itself).

Step 2: Apply SAS congruence

By the SAS congruence postulate, the two triangles are congruent.

Problem 10:

Step 1: Identify the sides and angle

In problem 10, we have two right - angled triangles. We can see that the legs of the right - angled triangles: one leg has two tick marks, another leg has one tick mark, and the right angle (included angle) is congruent (since all right angles are congruent, $90^{\circ}$).

Step 2: Apply SAS congruence

By the SAS congruence postulate, since two sides (the legs) and the included angle (the right angle) of one right - angled triangle are congruent to two sides (the legs) and the included angle (the right angle) of the other right - angled triangle, the two triangles are congruent.

Summary of congruence for each problem (using SAS):

1. Congruent (SAS, common side, congruent sides, congruent included angle)
2. Congruent (SAS, common side, congruent sides, congruent included angle)
3. Congruent (SAS, common side, congruent sides, congruent included angle)
4. Congruent (SAS, congruent sides, congruent included angle (right angle), common side)
5. Congruent (SAS, common side, congruent sides, congruent included angle)
6. Congruent (SAS, common side, congruent sides, congruent included angle)
7. Not congruent (SAS not satisfied)
8. Congruent (SAS, common side, congruent sides, congruent included angle)
9. Congruent (SAS, common side, congruent sides, congruent included angle)
10. Congruent (SAS, congruent legs, congruent right angle, congruent other leg)

If you want a more detailed answer for a specific problem, please let me know which one (e.g., problem 1, problem 5, etc.), and I can provide a more in - depth step - by - step for that particular problem.

Final Answer (for problem 1 as an example):

The two triangles are congruent by the SAS (Side - Angle - Side) congruence postulate.