Angle-Angle Similarity Postulate
If two angles of one triangle are similar to two angles of another triangle, the two triangles are similar.
In the example, angle E is congruent to angle A, and angle D is congruent to angle R. Since the two pairs of angles are congruent, triangle EDW is similar to triangle ARO by the Angle-Angle Similarity Postulate.
I chose this postulate because it helps prove triangles similar. Triangle similarity is important because we can use the things we learned in the previous lessons (proportions and similarity) and apply them to similar triangles. |
Side-Side-Side Proportionality Theorem
If three sides of one triangle are proportional to three sides of another triangle, the two triangles are similar.
All three sides of triangle ABC are proportional to all three sides of triangle DEF. 4/16 =1/4. 5/20 = 1/4. 6/24 = 1/4. Therefore, triangle ABC is similar to triangle DEF by SSS Prop. Theorem.
I chose the Side-Side-Side Proportionality Theorem because it is one of the three ways to prove triangles similar. In my opinion, it is the easiest way because you don't need to solve for any side lengths after you prove SSS.
I chose the Side-Side-Side Proportionality Theorem because it is one of the three ways to prove triangles similar. In my opinion, it is the easiest way because you don't need to solve for any side lengths after you prove SSS.
Side-Angle-Side Proportionality Theorem
If two sides of one triangle are proportional to those another triangle and their included angle is congruent, the two triangles are similar.
In the diagram, we can use the Side-Angle-Side Proportionality Theorem to prove triangle ABE is similar to triangle ACD. Side AB (3) divided by side AC (9) is 1/3. Side AE divided by side AD is also equal to 1/3. These two pairs of sides are proportional. Angle A is congruent to angle A by the reflexive property. The two sides of the triangles are proportional and their included angle is congruent, so ABE is similar to ACD by the SAS Prop. Theorem.
I chose the SAS Proportionality Theorem because it is another common way to prove triangles to be similar.
I chose the SAS Proportionality Theorem because it is another common way to prove triangles to be similar.
Triangle Proportionality Theorem
If a line is parallel is to a side of a triangle and intersects the other two sides, it divides those sides proportionally.
BE is parallel to CD and intersects sides AC and AD. Because of this, we can use the Triangle Proportionality Theorem. Since BE divides the two sides proportionally, x1 divided by x2 equals y1 divided by y2.
I chose the Triangle Proportionality Theorem because it expands on the things we learned in lesson 1 and 2 and helps us find the lengths of missing sides.
I chose the Triangle Proportionality Theorem because it expands on the things we learned in lesson 1 and 2 and helps us find the lengths of missing sides.
Triangle Angle Bisector Theorem
If a line is an angle bisector of a triangle, it divides the opposite side into two segments whose lengths are proportional to the lengths of the other two sides.
BD is the angle bisector of angle B. This means that the segments it creates side are proportional to the lengths of the other two sides. x1/x2 would get you the same ratio as y1/y2. x1/y1 would also get you the same ratio as x2/y2.
I chose this theorem because it is a way to solve for the lengths of sides when there is an angle bisector. The more ways we know, the more prepared we are for different types of problems.
I chose this theorem because it is a way to solve for the lengths of sides when there is an angle bisector. The more ways we know, the more prepared we are for different types of problems.
Similarity in Right Triangles Theorem & Corollaries
The altitude to the hypotenuse of a right triangle forms two triangles that are similar to each other and to the
original triangle.
original triangle.
BD is the altitude of triangle ABC. This means that it creates two similar right triangles, ADB and BDC. The altitude is the geometric mean of the two segments on the hypotenuse. The length of a leg is the geometric mean of the lengths of the hypotenuse and the segment adjacent to that leg.
This theorem is important because many corollaries are created off of it. (Heartbeat & Rabbit) It allows us to solve for missing sides when there is an altitude going from the right angle to the hypotenuse.
This theorem is important because many corollaries are created off of it. (Heartbeat & Rabbit) It allows us to solve for missing sides when there is an altitude going from the right angle to the hypotenuse.
Practice Problems: https://docs.google.com/a/k12.friscoisd.org/presentation/d/1Ym-lZ_bhgX9pImoTAukeTZWPf_O7sTWgtBNZEpQdinM/edit#slide=id.p