Triangles P Q R, P prime q prime R prime, and S T U. Angles Q and R are congruent. Side Q R is 2 and R P is 3. Angles Q prime and R prime are congruent. Side Q prime R prime if 1 and R prime and P prime is 1.5. Side T U is 1 and S U is 1 point 5.
Triangle is congruent to triangle .
Triangle is similar to triangle .
Triangles G H I, G prime H prime I prime, and M N O. Angle G corresponds to angle G prime and angle M. Angle H corresponds to angle H prime and angle N. Side G H is congruent to side M N.
Triangle is congruent to triangle .
Triangle is similar to triangle .
9.2
Activity
How Many Pieces?
For each problem, draw 2 triangles that have the listed properties. Try to make them as different as possible.
One angle is 45 degrees.
One angle is 45 degrees, and another angle is 30 degrees.
One angle is 45 degrees, and another angle is 30 degrees. The lengths of a pair of corresponding sides are 2 cm and 6 cm.
Compare your triangles with the other triangles from your group. Do any of the conditions guarantee that the triangles will be similar? Make a conjecture.
Prove your conjecture.
9.3
Activity
Any Two Angles?
Here are 2 triangles. One triangle has a 60 degree angle and a 40 degree angle. The other triangle has a 40 degree angle and an 80 degree angle.
2 triangles. One triangle has a 60 degree angle and a 40 degree angle. Side opposite 40 degree angle has a length of 3, side opposite 60 degrees has a length of x, and third side has a length of 6.
The second triangle has a 40 degree angle and an 80 degree angle. Side opposite 40 degree angle has a length of 5, side opposite 80 degrees has a length of y, and third side has a length of 8.
How can you show that the triangles are similar?
How long are the sides labeled and ?
Student Lesson Summary
We already know that when two figures are congruent there is a sequence of rigid motions that takes one figure onto the other. So, if a dilation takes Figure A to an image that is congruent to Figure B, then Figure A and Figure B are similar because there is a sequence of a dilation and rigid motions that takes Figure A onto Figure B.
We can use this idea to show that when two angles of one triangle are congruent to two angles of a second triangle, then the two triangles are similar. We call this the Angle-Angle Triangle Similarity Theorem.
In the diagram, angle is congruent to angle , and angle is congruent to angle . If a sequence of rigid motions and dilations moves the first figure so that it fits exactly over the second, then we have shown that the Angle-Angle Triangle Similarity Theorem is true.
Dilate triangle by the ratio , so that is congruent to . Now triangle is congruent to triangle by the Angle-Side-Angle Triangle Congruence Theorem, which means that there is a sequence of rotations, reflections, and translations that takes onto .
Two triangles A B C and D E F. Point C prime lies on side A C and point B prime lies on side A B. A line segment is drawn from point C prime to point B prime forming triangle A prime B prime C prime. Angle A prime corresponds to angle D, angle B prime corresponds to angle E, and angle B corresponds to angle E.
Therefore, a dilation followed by a sequence of rotations, reflections, and translations will take triangle onto triangle , which is the definition of similarity. We have shown that a dilation and a sequence of rigid motions takes triangle to triangle , so the triangles are similar.
