Review of Geometry Prepared by Title V Staff: Daniel Judge, Instructor Ken Saita, Program Specialist East Los Angeles College EXIT TOPICSBACKNEXT © 2002 East Los Angeles College. All rights reserved. Click one of the buttons below or press the enter key

Topics Lines Angles Triangles Click on the topic that you wish to view... EXIT TOPICSBACKNEXT

Lines EXIT TOPICSBACKNEXT

When a pair of lines are drawn, the portion of the plane where the lines do not intersect is divided into three distinct regions. Region 1 Region 3 Region 2 EXIT TOPICSBACKNEXT

These regions are referred to as: Interior Region – Region bounded by both lines. Exterior Region – The remaining outside regions. exterior interior EXIT TOPICSBACKNEXT

Parallel Lines – Lines that never intersect. l1l1 l2l2 Notation l 1 l 2 EXIT TOPICSBACKNEXT

Transversal – A line that intersects two or more lines in different points. l1l1 l2l2 Note: l 1 is not parallel to l 2 ( l 1 l 2 ) EXIT TOPICSBACKNEXT

Transversal l1l1 l2l2 Note: l 1 is parallel to l 2 ( l 1 l 2 ) EXIT TOPICSBACKNEXT

Angles EXIT TOPICSBACKNEXT

Angles are formed when lines intersect. l1l1 l2l2 Note: ( l 1 l 2 ) A B C D EXIT TOPICSBACKNEXT

 A and  B are said to be adjacent. (neighbors) l1l1 l2l2 A B C D EXIT TOPICSBACKNEXT

l1l1 l2l2 A B C D Adjacent Angles – Angles that share a common vertex and a common side between them. EXIT TOPICSBACKNEXT

l1l1 l2l2 A B C D Note:  B and  C are adjacent (neighbors)  C and  D are adjacent (neighbors)  D and  A are adjacent (neighbors) EXIT TOPICSBACKNEXT

l1l1 l2l2 A B C D Vertical Angles – The pairs of non-adjacent angles formed by the intersection of two lines. EXIT TOPICSBACKNEXT

l1l1 l2l2 A B C D Note:  A and  C are vertical angles  B and  D are vertical angles EXIT TOPICSBACKNEXT

Q: What’s special about vertical angles? Answer – They have the same measure. (they are congruent) l1l1 l2l2 110° 70° EXIT TOPICSBACKNEXT

Fact – When you intersect two lines at a point l1l1 l2l2 A C BD  A   C (congruent)  B   D (congruent) EXIT TOPICSBACKNEXT

Two angles are said to be supplementary if their sum measures 180°. Adjacent angles formed by two intersecting lines are supplementary. l1l1 l2l2 A C BD  A and  B are supplementary angles. EXIT TOPICSBACKNEXT

Can you find any other supplementary angles in the figure below? l1l1 l2l2 A C BD EXIT TOPICSBACKNEXT

Note: Angles whose sum measures 90° are said to be complementary. EXIT TOPICSBACKNEXT

Revisiting the transversal, copy this picture in your notebook. l1l1 l2l2 Note: ( l 1 l 2 ) AB C D H G E F EXIT TOPICSBACKNEXT

Angles in the interior region between the two lines are called interior angles. Angles in the exterior region are called exterior angles. l1l1 l2l2 AB C D H G E F Interior Exterior EXIT TOPICSBACKNEXT

Q: Which are the interior angles and exterior angles? l1l1 l2l2 AB C D H G E F EXIT TOPICSBACKNEXT

l1l1 l2l2 AB C D H G E F Answer— InteriorExterior  C  A  D  B  E  G  F  H EXIT TOPICSBACKNEXT

Q: Which angles are adjacent? Q: Which angles are vertical? Q: Which angles are supplementary? l1l1 l2l2 AB C D H G E F EXIT TOPICSBACKNEXT

Consider a transversal consisting of the two parallel lines. l1l1 l2l2 A C B D FE GH EXIT TOPICSBACKNEXT

l1l1 l2l2 A C B D FE GH We know,  A   D  B   C  E   H  G   F since they are all vertical angles. EXIT TOPICSBACKNEXT

Q: Are any other angles congruent? EXIT TOPICSBACKNEXT

Yes! If we could slide l 2 up to l 1, we would be looking at the following picture. EXIT TOPICSBACKNEXT

l1l1 l2l2 A C B D FE GH This means the following is true:  A and  E have the same measure (congruent)  B and  F have the same measure (congruent)  C and  G have the same measure (congruent)  D and  H have the same measure (congruent) EXIT TOPICSBACKNEXT

Having knowledge of one angle in the special transversal below, allows us to deduce the rest of the angles. l1l1 l2l2 120° C B D FE GH l 1 l 2 What are the measures of the other angles? EXIT TOPICSBACKNEXT

Answer: l1l1 l2l2 120°60° l 1 l 2 60° 120° 60° 120° Why? EXIT TOPICSBACKNEXT

Triangles EXIT TOPICSBACKNEXT

One of the most familiar geometric objects is the triangle. In fact, trigonometry is the study of triangles EXIT TOPICSBACKNEXT

Triangles have two important properties 1. 3 sides 2. 3 interior angles A BC EXIT TOPICSBACKNEXT

We also have some special triangles. EXIT TOPICSBACKNEXT

Right Triangle — One interior angle of the triangle measures 90° (has a right angle) EXIT TOPICSBACKNEXT

Equilateral Triangle — 1. All of the sides are congruent (have the same measure). EXIT TOPICSBACKNEXT

Equiangular Triangle — 1. All of the interior angles are congruent (have the same measure). EXIT TOPICSBACKNEXT

Note – Equiangular triangles are also equilateral triangles. Equilateral triangles are also equiangular triangles. EXIT TOPICSBACKNEXT

Isosceles Triangle — 1. Two of the interior angles of the triangle are congruent (have the same measure). 2. Two of the sides are congruent. EXIT TOPICSBACKNEXT

The sum of the interior angles of any triangle measures 180° A BC That is,  A +  B +  C = 180° EXIT TOPICSBACKNEXT

Why? EXIT TOPICSBACKNEXT

Form a transversal with two parallel lines. A BC EXIT TOPICSBACKNEXT

Fill in the missing vertical angles. A BC EXIT TOPICSBACKNEXT

Solution-- A BC A BC EXIT TOPICSBACKNEXT

Fill in the remaining angles. A BC A BC EXIT TOPICSBACKNEXT

Solution-- A BC A BC Do you notice anything? BC EXIT TOPICSBACKNEXT

That is,  B +  A +  C = 180° A BC A BC Note – The order in which we add doesn’t matter. BC EXIT TOPICSBACKNEXT

A BC  A +  B +  C = 180° (This is true for any triangle) EXIT TOPICSBACKNEXT

End of Review of Geometry Title V East Los Angeles College 1301 Avenida Cesar Chavez Monterey Park, CA Phone: (323) Us At: Our Website: EXIT TOPICSBACKNEXT