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Course: Digital SAT Math > Unit 13

Lesson 2: Congruence, similarity, and angle relationships: advanced
Test prep
Digital SAT Math
Advanced: Geometry and trigonometry
Congruence, similarity, and angle relationships: advanced
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Congruence, similarity, and angle relationships | Lesson

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A guide to congruence, similarity, and angle relationships on the digital SAT

What are congruence, similarity, and angle relationship problems?

Congruence, similarity, and angle relationship problems ask us to solve for an unknown value using
,
(shown below), and intersections of lines.
Triangles A B C and X Y Z. The triangles are different sizes. But angle A is congruent to angle X. Angle B is congruent to angle Y, and angle C is congruent to angle Z.
You can learn anything. Let's do this!

What are some common ways the SAT combines angle relationships?

Finding angles in triangles

Khan Academy video wrapper
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.
To solve for unknown angle measures, we need to know the following information:
  • 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.
Three lines intersect to form a triangle. The angles of the triangle measure x degrees, y degrees, and z degrees.
We know that x∘+y∘+z∘=180∘, but we also know how the angles outside the triangle relate to the inside angles based on the properties of
and
.
At the vertex where the x degree triangle angle is, the vertical angle measures x degrees, and the adjacent angles measure 180 degrees minus x degrees. At the vertex where the y degree triangle angle is, the vertical angle measures y degrees, and the adjacent angles measure 180 degrees minus y degrees. At the vertex where the z degree triangle angle is, the vertical angle measures z degrees, and the adjacent angles measure 180 degrees minus z degrees.

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.
Two horizontal lines and two transversals create two similar triangles between the two horizontal lines. In the bottom triangle, the bottom left angle measures x degrees, the top angle measures y degrees, and the bottom right angle measures z degrees. In the top triangle, the top left angle measures z degrees, the bottom angle measures y degrees, and the top right angle measures x degrees.
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.
A triangle has a horizontal base. Its bottom left angle measures x degrees, its top angle measures y degrees, and its bottom right angle measures z degrees. A horizontal line inside the triangle divides the larger triangle into a smaller triangle on top and a quadrilateral below. The smaller triangle shares its top angle with the larger triangle. The smaller triangle's bottom left angle measures x degrees, and its bottom right angle measures z degrees.

Try it!

try: find angle measures in an intersection of three lines
Three lines intersect to form a triangle. The left angle of the triangle measures 60 degrees, the top angle of the triangle measures 75 degrees, and the bottom-right angle of the triangle measures a degrees. The angle opposite of the 60-degree angle measure b degrees, and the angle adjacent to the 75-degree angle measures c degrees.
Based on the figure above, what is the value of a ?
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi
What is the value of b ?
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi
What is the value of c ?
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi

try: find angle measures of parallel lines and transversals
Two horizontal lines and two transversals create two similar triangles between the two horizontal lines. In the bottom triangle, the bottom left angle measures x degrees, the top angle measures y degrees, and the bottom right angle measures z degrees. In the top triangle, the top left angle measures z degrees, the bottom angle measures y degrees, and the top right angle measures x degrees.
The figure above shows two horizontal lines and two intersecting transversals.
What is the value of a ?
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi
What is the value of b ?
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi
What is the value of c ?
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi

How do I use similarity to find side lengths?

Solving similar triangles

Khan Academy video wrapper
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 ABC and XYZ are similar: they have the same angle measures, but not the same side lengths.
Triangles A B C and X Y Z. The triangles are different sizes. But angle A is congruent to angle X. Angle B is congruent to angle Y, and angle C is congruent to angle Z.
The corresponding side lengths of similar triangles are related by a constant ratio, which we can call k. For similar triangles ABC and XYZ, the following is true:
XY=k(AB)YZ=k(BC)XZ=k(AC)XYAB=YZBC=XZAC=k
Let's try applying the properties of similar triangles. In the figure below, BD― is parallel to AE―. If BC=10, BD=14, and AE=21, what is the length of AC― ?
Triangle ACE has horizontal base AE with length 21 and top vertex C. B is a point on side AC, and D is a point on side CE. Line segment BD has length 14, and line segment BC has length 10.

Try it!

try: use similarity to find side length
Triangles A B D and B C D share side B D. Triangle A B D has a right angle at vertex D. Side A D of triangle A B D is collinear with side C D of triangle B C D.
In the figure above, triangles ABD and BCD are similar. The length of BD― is 6, and the length of CD― is 12.
Based on the figure, CD― is the
of triangle BCD, and BD― is both the
of triangle BCD and the
of triangle ABD.
What is the length of AD― ?
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi

Your turn!

practice: find an angle measure
Lines p, q, and r intersect to form a triangle. The top right angle of the triangle measures 24 degrees. The top left angle of the triangle and a 139-degree angle are supplementary angles. The bottom angle of the triangle and an angle measuring x degrees are vertical angles.
Intersecting lines p, q, and r are shown above. What is the value of x ?
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi

practice: find an angle measure
Lines l and m are parallel. Line m is collinear with the base of a triangle, and line l divides the larger triangle into a smaller triangle and a trapezoid. The two triangles share the same top angle, which measures y degrees. The bottom left angle of the smaller triangle measures x degrees, and the bottom right angle of the larger triangle measures z degrees.
In the figure above, lines ℓ and m are parallel, y=30, and z=45. What is the value of x ?
Choose 1 answer:

practice: find a side length
AB is a vertical line segment of length 25. DE is a vertical line segment with length 50. Line segments AD and BE intersect at point C between the two vertical line segments. The length of line segment BD is 20.
In the figure above, segments AB and DE are parallel, and segments AD and BE intersect at C. What is the length of segment BE ?
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi

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