PARALLEL LINES CUT BY A TRANSVERSAL DEFINITIONS PARALLEL

PARALLEL LINES CUT BY A TRANSVERSAL

DEFINITIONS • • PARALLEL TRANSVERSAL ANGLE VERTICAL ANGLE CORRESPONDING ANGLE ALTERNATE INTERIOR ANGLE ALTERNATE EXTERIOR ANGLE

DEFINITIONS • SUPPLEMENTARY ANGLE • COMPLEMENTARY ANGLE • CONGRUENT

Parallel lines cut by a transversal 2 3 7 6 8 5 1 4

Parallel lines cut by a transversal 2 3 7 6 8 1 4 5 < 1 and < 2 are called SUPPLEMENTARY ANGLES DEFINITION: They form a straight angle measuring 180 degrees.

Parallel lines cut by a transversal 2 3 7 6 8 1 4 5 Name other supplementary pairs: < 2 and < 3 and < 4 and < 1 < 5 and < 6 and < 7 and < 8 and < 5

Parallel lines cut by a transversal 2 3 7 6 8 1 4 5 < 1 and < 3 are called VERTICAL ANGLES They are congruent m<1 = m<3 DEFINITION: The angles formed from two lines are crossing.

Parallel lines cut by a transversal 2 3 7 6 8 1 4 5 < 2 and < 4 < 6 and < 8 Name other vertical pairs: < 5 and < 7

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

Parallel lines cut by a transversal 2 3 7 6 8 1 4 5 < 2 and < 6 < 3 and < 7 Name other corresponding pairs: < 4 and < 8

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

Parallel lines cut by a transversal 2 3 7 6 8 1 4 5 Name other alternate interior pairs: < 3 and < 5

Parallel lines cut by a transversal 2 3 7 6 8 1 4 5 < 1 and < 7 are called ALTERNATE EXTERIOR ANGLES They are congruent m<1 = m<7 Alternate Exterior on the outside of the two parallel lines and on opposite sides of the transversal.

Parallel lines cut by a transversal 2 3 7 6 8 1 4 5 < 2 and < 8 Name other alternate exterior pairs: < 1 and < 7

Parallel lines cut by a transversal 2 3 7 6 8 1 4 5 < 4 and < 5 are called CONSECUTIVE INTERIOR ANGLES The sum is 180. m<4 = m<5 DEFINITION: Consecutive Interior on the inside of the two parallel lines and on same side of the transversal. Sum = 180

TRY IT OUT 2 3 7 The m < 1 is 6 8 4 5 60 degrees. What is the m<2 ? 1 120 degrees

TRY IT OUT 2 3 7 The m < 1 is 6 8 4 5 60 degrees. What is the m<5 ? 1 60 degrees

TRY IT OUT 2 3 7 The m < 1 is 6 8 4 5 60 degrees. What is the m<3 ? 1 60 degrees

TRY IT OUT 120 60 60 120

TRY IT OUT 2 x + 20 What do you know about the angles? Write the equation. Solve for x. x + 10 SUPPLEMENTARY 2 x + 20 + x + 10 = 180 3 x + 30 = 180 3 x = 150 x = 50

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

3 -2 Angles Formed by Parallel Lines and Transversals Warm Up Identify each angle pair. 1. 1 and 3 corr. s 2. 3 and 6 alt. int. s 3. 4 and 5 alt. ext. s 4. 6 and 7 same-side int s Holt Geometry

3 -2 Angles Formed by Parallel Lines and Transversals Example 1: Using the Corresponding Angles Postulate Find each angle measure. A. m ECF x = 70 Corr. s Post. m ECF = 70° B. m DCE 5 x = 4 x + 22 x = 22 m DCE = 5 x = 5(22) = 110° Holt Geometry Corr. s Post. Subtract 4 x from both sides. Substitute 22 for x.

3 -2 Angles Formed by Parallel Lines and Transversals Check It Out! Example 1 Find m QRS. x = 118 Corr. s Post. m QRS + x = 180° m QRS = 180° – x Def. of Linear Pair Subtract x from both sides. = 180° – 118° Substitute 118° for x. = 62° Holt Geometry

WEBSITES FOR PRACTICE Ask Dr. Math: Corresponding /Alternate Angles Project Interactive: Parallel Lines cut by Transversal

Triangle Sum Theorem The sum of the angle measures in a triangle equal 180° 1 3 2 m<1 + m<2 + m<3 = 180°

Corollary • A corollary to a theorem is a statement that follows directly from that theorem

TRIANGLE SUM THEOREM COROLLARIES • If 2 angles of 1 triangle are congruent to 2 angles of another triangle, then the 3 rd angles are congruent • The acute angles of a right triangle are complementary • The measure of each angle of an equiangular triangle is 60 o • A triangle can have at most 1 right or 1 obtuse angle

Exterior Angle Theorem (your new best friend) The measure of an exterior angle in a triangle is the sum of the measures of the 2 remote interior angles exterior angle 2 1 3 m<4 = m<1 + m<2 4

REMOTE INTERIOR ANGLE • In any polygon, a remote interior angle is an interior angle that is not adjacent to a given exterior angle A and B are remote to angle 1

Exterior Angle Theorem interior In the triangle below, recall that � 1, � 2, and � 3 are _______ angles of ΔPQR. Angle 4 is called an exterior _______ angle of ΔPQR. linear pair with one of An exterior angle of a triangle is an angle that forms a _____ the angles of the triangle. In ΔPQR, � 4 is an exterior angle at R because it forms a linear pair with � 3. Remote interior angles of a triangle are the two angles that do not form __________ a linear pair with the exterior angle. In ΔPQR, � 1, and � 2 are the remote interior angles with respect to � 4. P 1 Q 2 3 4 R

Exterior Angle Theorem In the figure below, � 2 and � 3 are remote interior angles with respect to what angle? � 5 1 2 3 4 5

an example with numbers find x & y 82° 30° x y x = 68° y = 112° • Determine the measure of <4, 30 x find all the angle measures • If <3 = 50, <2 = 70 40 x 10 x 2 80°, 60°, 40° Do you hear the sirens? ? ?

Exterior Angle Theorem

4 -2 Angle Relationships in Triangles Example 3: Applying the Exterior Angle Theorem Find m B. m A + m B = m BCD Ext. Thm. 15 + 2 x + 3 = 5 x – 60 Substitute 15 for m A, 2 x + 3 for m B, and 5 x – 60 for m BCD. 2 x + 18 = 5 x – 60 78 = 3 x Simplify. Subtract 2 x and add 60 to both sides. Divide by 3. 26 = x m B = 2 x + 3 = 2(26) + 3 = 55° Holt Geometry

4 -2 Angle Relationships in Triangles Holt Geometry

There are several ways to prove certain triangles are similar. The following postulate, as well as the SSS and SAS Similarity Theorems, will be used in proofs just as SSS, SAS, ASA, HL, and AAS were used to prove triangles congruent.

Example 1: Using the AA Similarity Postulate Explain why the triangles are similar and write a similarity statement. Since , B E by the Alternate Interior Angles Theorem. Also, A D by the Right Angle Congruence Theorem. Therefore ∆ABC ~ ∆DEC by AA~.

Check It Out! Example 1 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 ~.