Angles with Parallel Lines Topic Index Geometry Index Regents Exam Prep Center

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1 Angles with Parallel Lines Topic Index Geometry Index Regents Exam Prep Center A transversal is a line that intersects two or more lines (in the same plane). When lines intersect, angles are formed in several locations. Certain angles are given "names" that describe "where" the angles are located in relation to the lines. These names describe angles whether the lines involved are parallel or not parallel. Remember that: - the word INTERIOR means BETWEEN the lines. - the word EXTERIOR means OUTSIDE the lines. - the word ALTERNATE means "alternating sides" of the transversal. When the lines are NOT parallel... When the lines are parallel...

2 The names "alternate interior angles", "alternate exterior angles", "corresponding angles", and "interior angles on the same side of the transversal" are used to describe specific angles formed when lines intersect. These names are used both when lines are parallel and when lines are not parallel. Let's examine these angles, and other angles, when the lines are parallel. When the lines are parallel: Alternate Interior Angles (measures are equal) The name clearly describes "where" these angles are located. Look carefully at the diagram below:

3 Hint: If you draw a Z on the diagram, the alternate interior angles are found in the corners of the Z. The Z may also be a backward Z. Theorem: If two parallel lines are cut by a transversal, the alternate interior angles are congruent. Theorem: If two lines are cut by a transversal and the alternate interior angles are congruent, the lines are parallel. When the lines are parallel: Alternate Exterior Angles (measures are equal) The name clearly describes "where" these angles are located. Look carefully at the diagram below:

4 Theorem: If two parallel lines are cut by a transversal, the alternate exterior angles are congruent. Theorem: If two lines are cut by a transversal and the alternate exterior angles are congruent, the lines are parallel. When the lines are parallel: Corresponding Angles (measures are equal) Unfortunately, the name of these angles does not clearly indicate "where" they are located. They are located: - on the SAME SIDE of the transversal - one INTERIOR and one EXTERIOR - and they are NOT adjacent (they don't touch). (They lie on the same side of the transversal, in corresponding positions.)

5 Hint: If you took a picture of one corresponding angle and slid the angle up (or down) the same side of the transversal, you would arrive at the other corresponding angle. Also: If you draw an F on the diagram, the corresponding angles can be found in the "corners" of the F. The F may be backward and/or upside-down. Theorem: If two parallel lines are cut by a transversal, the corresponding angles are congruent. Theorem: If two lines are cut by a transversal and the corresponding angles are congruent, the lines are parallel.

6 When the lines are parallel: Interior Angles on the Same Side of the Transversal (measures are supplementary) Their "name" is simply a description of where the angles are located. Theorem: If two parallel lines are cut by a transversal, the interior angles on the same side of the transversal are supplementary. Theorem: If two lines are cut by a transversal and the interior angles on the same side of the transversal are supplementary, the lines are parallel. Of course, there are also other angle relationships occurring when working with parallel lines.

7 Vertical Angles (measures are equal) Vertical angles are ALWAYS equal, whether you have parallel lines or not. Refresh your memory using the diagram below: Theorem: Vertical angles are congruent. Angles forming a Linear Pair (Adjacent Angles creating a Straight Line) (measures are supplementary) This is an "old" idea about angles revisited. Since a straight angle contains 180, these two adjacent angles add to 180. They form a linear pair. (Adjacent angles share a vertex, share a side, and do not overlap.)

8 Theorem: If two angles form a linear pair, they are supplementary. Congruent Triangles Topic Index Geometry Index Regents Exam Prep Center Created by Donna Roberts Copyright Oswego City School District Regents Exam Prep Center (see Congruent for more info) When two triangles are congruent they will have exactly the same three sides and exactly the same three angles. The equal sides and angles may not be in the same position (if there is a turn or a flip), but they are there. Same Sides When the sides are the same then the triangles are congruent. For example:

9 is congruent to: and because they all have exactly the same sides. But: is NOT congruent to: because the two triangles do not have exactly the same sides. Same Angles Does this also work with angles? Not always! Two triangles with the same angles might be congruent: is congruent to: only because they are the same size But they might NOT be congruent because of different sizes:

10 is NOT congruent to: because, even though all angles match, one is larger than the other. So just having the same angles is no guarantee they are congruent. Other Combinations There are other combinations of sides and angles that can work read more at How To Find if Triangles are Congruent Marking When two triangles are congruent we often mark corresponding sides and angles like this: is congruent to: The sides marked with one line are equal in length. Similarly for the sides marked with two lines. Also for the sides marked with three lines. The angles marked with one arc are equal in size. Similarly for the angles marked with two arcs. Also for the angles marked with three arcs. How To Find if Triangles are Congruent Two triangles are congruent if they have: exactly the same three sides and exactly the same three angles. But we don't have to know all three sides and all three angles...usually three out of the six is enough.

11 There are five ways to find if two triangles are congruent: SSS, SAS, ASA, AAS and HL. 1. SSS (side, side, side) SSS stands for "side, side, side" and means that we have two triangles with all three sides equal. For example: is congruent to: (See Solving SSS Triangles to find out more) If three sides of one triangle are equal to three sides of another triangle, the triangles are congruent. 2. SAS (side, angle, side) SAS stands for "side, angle, side" and means that we have two triangles where we know two sides and the included angle are equal. For example:

12 is congruent to: (See Solving SAS Triangles to find out more) If two sides and the included angle of one triangle are equal to the corresponding sides and angle of another triangle, the triangles are congruent. 3. ASA (angle, side, angle) ASA stands for "angle, side, angle" and means that we have two triangles where we know two angles and the included side are equal. For example: is congruent to: (See Solving ASA Triangles to find out more) If two angles and the included side of one triangle are equal to the corresponding angles and side of another triangle, the triangles are congruent. 4. AAS (angle, angle, side)

13 AAS stands for "angle, angle, side" and means that we have two triangles where we know two angles and the non-included side are equal. For example: is congruent to: (See Solving AAS Triangles to find out more) If two angles and the non-included side of one triangle are equal to the corresponding angles and side of another triangle, the triangles are congruent. 5. HL (hypotenuse, leg) This one applies only to right angled-triangles! or HL stands for "Hypotenuse, Leg" (the longest side of a right-angled triangle is called the "hypotenuse", the other two sides are called "legs") It means we have two right-angled triangles with

14 the same length of hypotenuse and the same length for one of the other two legs. It doesn't matter which leg since the triangles could be rotated. For example: is congruent to: (See Pythagoras' Theorem to find out more) If the hypotenuse and one leg of one right-angled triangle are equal to the corresponding hypotenuse and leg of another right-angled triangle, the two triangles are congruent. Caution! Don't Use "AAA"! AAA means we are given all three angles of a triangle, but no sides. This is not enough information to decide if two triangles are congruent! Because the triangles can have the same angles but be different sizes:

15 is not congruent to: Without knowing at least one side, we can't be sure if two triangles are congruent.

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