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SAT

Course: SAT>Unit 6

Lesson 6: Additional Topics in Math: lessons by skill

Congruence and similarity | Lesson

What are congruence and similarity problems, and how frequently do they appear on the test?

Congruence and similarity problems ask us to use
and
(shown below) to solve problems.
On your official SAT, you'll likely see 1 to 2 questions about congruence and similarity.
You can learn anything. Let's do this!

What are some common ways the SAT combines angle relationships?

Finding angles in triangles

Worked example: Triangle angles (intersecting lines)See video transcript

Triangles and other angle relationships

On the SAT, we're expected to find unknown angle measures when only a few are given. More often than not, triangles are involved.
At the beginning of each SAT math section, the following information is provided as reference:
• The sum of the measures in degrees of the angles of a triangle is 180.

Triangles, vertical angles, and supplementary angles

One common type of figure on the SAT is a triangle formed by three intersecting lines, as shown below.
We know that start color #7854ab, x, degrees, end color #7854ab, plus, start color #ca337c, y, degrees, end color #ca337c, plus, start color #208170, z, degrees, end color #208170, equals, 180, degrees, but we also know how the angles outside the triangle relate to the inside angles based on the properties of
and
.

Triangles and parallel lines

Another common type of figure shows
constructed using parallel lines.
Two similar triangles can be constructed from two parallel lines and two intersecting transversals, as shown below.
Note: Since the two triangles have different orientations, be careful when identifying the corresponding sides! In two similar triangles, the longest side in one corresponds to the longest side in the other and so on.
Two similar triangles can also be constructed by drawing a line inside a triangle that's parallel to one of the sides. In the example shown below, the line inside the triangle is parallel to the base of the triangle and divides the larger triangle into a similar smaller triangle and a quadrilateral.

Try it!

try: find angle measures in an intersection of three lines
Based on the figure above, what is the value of a ?
What is the value of b ?
What is the value of c ?

try: find angle measures of parallel lines and transversals
The figure above shows two horizontal lines and two intersecting transversals.
What is the value of a ?
What is the value of b ?
What is the value of c ?

How do I use similarity to find side lengths?

Solving similar triangles

Solving similar trianglesSee video transcript

Setting up proportional relationships using similarity

Similar triangles have the same shape, but aren't necessarily the same size. In the figure below, triangles start color #7854ab, A, B, C, end color #7854ab and start color #ca337c, X, Y, Z, end color #ca337c are similar: they have the same angle measures, but not the same side lengths.
The corresponding side lengths of similar triangles are related by a constant ratio, which we can call k. For similar triangles start color #7854ab, A, B, C, end color #7854ab and start color #ca337c, X, Y, Z, end color #ca337c, the following is true:
\begin{aligned} \maroonD{XY}&=k(\purpleD{AB}) \\\\ \maroonD{YZ} &=k(\purpleD{BC}) \\\\ \maroonD{XZ} &=k(\purpleD{AC}) \\\\ \dfrac{\maroonD{XY}}{\purpleD{AB}}&=\dfrac{\maroonD{YZ}}{\purpleD{BC}}=\dfrac{\maroonD{XZ}}{\purpleD{AC}}=k \end{aligned}
Let's try applying the properties of similar triangles. In the figure below, start overline, B, D, end overline is parallel to start overline, A, E, end overline. If B, C, equals, 10, B, D, equals, 14, and A, E, equals, 21, what is the length of start overline, A, C, end overline ?

Try it!

try: use similarity to find side length
In the figure above, triangles A, B, D and B, C, D are similar. The length of start overline, B, D, end overline is 6, and the length of start overline, C, D, end overline is 12.
Based on the figure, start overline, C, D, end overline is the
of triangle B, C, D, and start overline, B, D, end overline is both the
of triangle B, C, D and the
of triangle A, B, D.
What is the length of start overline, A, D, end overline ?

practice: find an angle measure
Intersecting lines p, q, and r are shown above. What is the value of x ?

practice: find an angle measure
In the figure above, lines ell and m are parallel, y, equals, 30, and z, equals, 45. What is the value of x ?

practice: find a side length
In the figure above, segments A, B and D, E are parallel, and segments A, D and B, E intersect at C. What is the length of segment B, E ?

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