Question

The SAS, or side-angle-side, criterion states that two triangles are similar if two pairs of corresponding sides are in the same ratio and their included angles are congruent. You’ll use GeoGebra to demonstrate this. Use the GeoGebra activity to complete each step below. For help, watch these short videos about using GeoGebra to draw points, lines, and angles and use the measurement tools.

Part A
Create a random triangle, △ABC. Record the lengths of sides BC and CA and the measure of the included angle. Take a screenshot of your dilation, save it, and insert the image below the table.
∠BCA
side BC
side CA

Part B
Draw DE parallel to BC. You can draw DE any length and place it anywhere on the coordinate plane, but not on top of ABC.

Find and record the ratio, n, of the length of DE to the length of BC. Then, multiply the length of CA by n and record the resulting length.

Part C
Now you will attempt to copy your original triangle using only two of its sides and the included angle:

Using point E as the center, draw a circle with a radius equal to the length of n x CA, which you calculated in part B.
Using point E as the vertex and DE as one side of the angle, create an angle that is equal to the measure of ∠BCA. Draw ray ED'.
Locate the intersection of the ray and the circle, and label the point F.
Complete △DEF by drawing a polygon through points D, E, and F.
Take a screenshot of your results, save it, and insert the image below.

Part D
Record the measures of the angles of △ABC and △DEF.
Angle Measure
∠CAB
∠BCA
∠ABC
Angle Measure
∠DFE
∠FED
∠EDF

Part E
Using your results, draw a conclusion about the relationship between two triangles when two pairs of corresponding side lengths are proportional by the same ratio and their included angles are congruent.