Which pair of triangles can be proven congruent by SAS?

Question:Which pair of triangles can be proven congruent by SAS?

Answer: The pair of right triangles formed by drawing the altitude from the apex of the isosceles triangle (first image) can be proven congruent by SAS.

Explanation:

Step1: Identify the Triangles

In the first image, we have an isosceles triangle. An altitude is dropped from the apex to the base, creating two right triangles on either side.

Step2: Determine the Corresponding Parts

Each resulting right triangle shares the altitude (a common side) and has one half of the base as another side. The altitude is perpendicular to the base, forming a right angle that is included between these two sides.

Step3: Verify SAS Conditions

• Side: The altitude segment is common to both triangles.
• Angle (Included): The right angle formed where the altitude meets the base is the included angle.
• Side: Each half of the base is congruent if the original triangle is isosceles.

Thus, both pairs of triangles have two sides and the included angle congruent, satisfying the SAS criterion.

Extended Knowledge:

Properties of Isosceles Triangles

In an isosceles triangle, the altitude to the base is also a median and angle bisector. Therefore, it splits the base into two congruent segments and forms two congruent right triangles.

Other Uses of SAS

The SAS criterion is often used when constructing and verifying properties in geometric figures such as parallelograms, trapezoids, and kites, where certain pairs of triangles inside these figures can be easily shown to be congruent.