Similar Triangle Criteria MCC9-12.G.SRT.2 Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. MCC9-12.G.SRT.3 Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar EQ: How do I know which method to use to prove two triangles similar?

Warm Up 2 1 3 4 6 5 7 8 The m < 1 is 45 degrees, find the remaining angles.

Vocabulary Angle: Angles are created by two distinct rays that share a common endpoint (also known as a vertex). ∠ABC or ∠B denote angles with vertex B. Adjacent Angles: Angles in the same plane that have a common vertex and a common side, but no common interior points. Alternate Exterior Angles : Alternate exterior angles are pairs of angles formed when a third line (a transversal) crosses two other lines. These angles are on opposite sides of the transversal and are outside the other two lines. When the two other lines are parallel, the alternate exterior angles are equal. Alternate Interior Angles: Alternate interior angles are pairs of angles formed when a third line (a transversal) crosses two other lines. These angles are on opposite sides of the transversal and are in between the other two lines. When the two other lines are parallel, the alternate interior angles are equal.

Complementary Angles: Two angles whose sum is 90 degrees. Vocabulary Complementary Angles: Two angles whose sum is 90 degrees. Linear Pair: Adjacent, supplementary angles. Excluding their common side, a linear pair forms a straight line. Supplementary Angles: Two angles whose sum is 180 degrees. Transversal: A line that crosses two or more lines. Vertical Angles: Two nonadjacent angles formed by intersecting lines or segments. Also called opposite angles.

Parallel lines cut by a transversal 2 1 3 4 6 5 7 8 < 1 and < 2 are called SUPLEMENTARY ANGLES They are a linear pair. ALL linear pairs are supplementary (their measures add up to 180̊ ). Name other supplementary pairs:

Parallel lines cut by a transversal 2 1 3 4 6 5 7 8 < 1 and < 3 are called VERTICAL ANGLES They are congruent m<1 = m<3 Name other vertical pairs:

Parallel lines cut by a transversal 2 1 3 4 6 5 7 8 < 1 and < 5 are called CORRESPONDING ANGLES They are congruent m<1 = m<5 Corresponding angles occupy the same position on the top and bottom parallel lines. Name other corresponding pairs:

Parallel lines cut by a transversal 2 1 3 4 6 5 7 8 < 4 and < 6 are called ALTERNATE INTERIOR ANGLES They are congruent m<4 = m<6 Alternate Interior on the inside of the two parallel lines and on opposite sides of the transversal. Name other alternate interior angles.

Parallel lines cut by a transversal 2 1 3 4 6 5 7 8 < 4 and < 6 are called ALTERNATE EXTERIOR ANGLES They are congruent m<2 = m<8 Alternate Interior on the inside of the two parallel lines and on opposite sides of the transversal. Name other alternate exterior angles.

TRY IT OUT 2 1 3 4 6 5 7 8 The m < 6 is 125 degrees, Find the rest of the angles.

TRY IT OUT 2x + 20 x + 10 What do you know about the angles? Write the equation. Solve for x. SUPPLEMENTARY 2x + 20 + x + 10 = 180 3x + 30 = 180 3x = 150 x = 30

TRY IT OUT 3x - 120 2x - 60 What do you know about the angles? Write the equation. Solve for x. ALTERNATE INTERIOR 3x - 120 = 2x - 60 x = 60 Subtract 2x from both sides Add 120 to both sides

Definition of Similar Triangles Two triangles are called similar if they both have the same three angle measurements. The two triangles shown are similar. Similar triangles have the same shape but POSSIBLY different sizes. You can think of similar triangles as one triangle being a magnification of the other.

Similar Triangle Notation B C E D F The two triangles shown are similar because they have the same three angle measures. The symbol for similarity is Here we write: The order of the letters is important: corresponding letters should name congruent angles.

Proportions from Similar Triangles Suppose Then the sides of the triangles are proportional, which means: Notice that each ratio consists of corresponding segments. A B C D E F

Example Given that if the sides of the triangles are as marked in the figure, find the missing sides. First, we write: Then fill in the values: Then: A B C D E F 8 6 7 12 9 10.5

Let’s stress the order of the letters again Let’s stress the order of the letters again. When we write note that the first letters are A and D, and The second letters are B and E, and The third letters are C and F, and We can also write: A B C E D F

Proving Triangles are Similar using Angle Angle Similarity (AA) postulate B C P T D One way to prove that two triangles are similar, is to show that two pairs of angles have the same measure. In the figure, This is because the unmarked angles are forced to have the same measure because the three angles of any triangle always add up to This is called the Third Angle Theorem: If two angles of one triangle are congruent to two angles of another triangle, then the third angles are also congruent.

When angles are congruent When trying to show that two triangles are similar, there are some standard ways of establishing that a pair of angles (one from each triangle) have the same measure: They may be given to be congruent. They may be vertical angles. They may be the same angle (sometimes two triangles share an angle). They may be a special pair of angles (like alternate interior angles) related to parallel lines. They may be in the same triangle opposite congruent sides. There are numerous other ways of establishing a congruent pair of angles.

Example 1 In the figure, Show that First, note that because these are alternate interior angles. Also, because these are alternate interior angles too. This is enough to show the triangles are similar, but notice the remaining pair of angles are vertical. ││ A B C D E

Example 2 Explain why the triangles are similar and write a similarity statement. By the Triangle Sum Theorem, mC = 47°, so C  F. B  E by the Right Angle Congruence Theorem. Therefore, ∆ABC ~ ∆DEF by AA ~.

Example 3 In the figure, are right angles, and Find First note that since and since the triangles share angle C. Let x denote AB. Then: E 12 9 A x B 6 C

In addition to AA, triangles can also be proven similar using the following criteria:

Example 4 Verify that the triangles are similar. ∆PQR and ∆STU Therefore ∆PQR ~ ∆STU by SSS ~.

Example 5 Verify that the triangles are similar. ∆DEF and ∆HJK D  H by the Definition of Congruent Angles. Therefore ∆DEF ~ ∆HJK by SAS ~.

Example 6 Verify that ∆TXU ~ ∆VXW. TXU  VXW by the Vertical Angles Theorem. Therefore ∆TXU ~ ∆VXW by SAS ~.

Lesson Quiz 1. Explain why the triangles are similar and write a similarity statement. 2. Explain why the triangles are similar, then find BE and CD.