Unit 3 Terms

Converse of Same-Side Interior/ Exterior Angles Theorem

* U in F.U.Z angles ______>___> | | |_____>____> If 2 lines and a transversal form same-side interior angles that are supplementary, then the 2 lines are parallel. Ex: If measure of angle 1 and 2 =180, then the 2 horizontal lines in the picture are parallel.

Converse of Alternate Interior/ Exterior Angles Theorem

* Z in F.U.Z. angles <--<-- ....... / ..... / .. / -->---> If 2 lines and a transversal form alternate interior or exterior angles that are congruent, then the 2 lines are parallel. Ex: refer to picture

Perpendicular Lines

* intersect & form a right angle * __|__ means perpendicular to Ex: The two lines in the picture form a right angle and that is what makes them perpendicular.

Parallel Lines

* will never intersect * can lie in different planes * || means parallel to * ------>------- = indicate parallelism Ex: q || m <----->q <----->m

Converse of Corresponding Angles Theorem

*The F in F.U.Z angles ---->-----> | ---->--> | | If 2 lines and a transversal form corresponding angles that are congruent, then the lines are parallel. Ex: If angles 1 and 5 are congruent, then l || m. If angles 2 and 6 are congruent, , then l || m. If angles 3 and 7 are congruent, then l || m. If angles 4 and 8 are congruent, then l || m.

Corresponding Angles Theorem

*The F in F.U.Z angles ---->-----> | ---->--> | | If a transversal intersects 2 parallel lines, then corresponding angles are CONGRUENT. Ex: If l || m, then angles 1 and 5 are congruent, 2 and 6 are congruent, 3 and 7 are congruent, & 4 and 8 are congruent.

Same-Side Exterior Angles Postulate

*The U in F.U.Z angles ______>___> | | |_____>____> If a transversal intersects 2 parallel lines, then same-side exterior angles are SUPPLEMENTARY -consecutive interior angles Ex: In the picture, angles 5 and 8 are supplementary as well as angles 1 and 4 are supplementary. ------------------------------------------------------- If l || m, then angle 5 + angle 8 = 180 degrees and angle 1 + angle 5 = 180 degrees. /5 right below <-----/----->l ......./ <--/-----> m /8 right above 4 = left of 8 and 1 = left of 5.

Same-Side Interior Angles Postulate

*The U in F.U.Z angles ______>___> | | |_____>____> If a transversal intersects 2 parallel lines, then same-side interior angles are SUPPLEMENTARY -consecutive interior angles Ex: In the picture, angles 4 and 6 are supplementary as well as angles 3 and 5 are supplementary. ------------------------------------------------------- If l || m, then angle 3 + angle 6 = 180 degrees and angle 4 + angle 5 = 180 degrees. / <-----/----->l ......./ 3 is right above and 6 is right below-in lines <--/-----> m 4 = left of 3 and 5 = left of 6.

Alternate Exterior Angles Theorem

*The Z in F.U.Z angles <--<-- ....... / ..... / .. / -->---> If a transversal intersects 2 parallel lines, then alternate exterior angles are CONGRUENT. Ex: (refer to picture) Measure of angles 2 and 7 are congruent. Measure of angles 1 and 8 are congruent.

Alternate Interior Angles Theorem

*The Z in F.U.Z angles <--<-- ....... / ..... / .. / -->---> If a transversal intersects 2 parallel lines, then alternate interior angles are CONGRUENT. Ex: (refer to picture) Measure of angles 3 and 6 are congruent. Measure of angles 4 and 5 are congruent.

Angle Addition Postulate

-If point B is in the interior of angle AOC, the measure of angle AOB plus measure of angle BOC = measure of angle AOC. If you place 2 angles side by side, then the measure of the resulting angle will be equal to the sum of the two original measures. Ex: Angles TRQ (40) and TRS (83) are side by side. 123 is equal to the sum of those measures.

Segment Addition Postulate

-If point B lines on line segment AC, then AB + BC = AC. -the distance from 1 endpoint to the other is the sum of the distances from the middle point to both endpoints. If 3 points, A, B, & C, are colinear and B is between A and C, then AB + BC = AC. Ex: Segment AB = 15 and segment BC = 35. This makes segment AC= 15(AB) + 35(BC).

Distributive Property

-Use multiplication to distribute a to each term of the sum or difference Ex: a(b+c) = a(b + c) = ab + ac SUM a(b-c) = a(b - c) = ab - ac DIFFERENCE

Definition of Supplementary Angles

2 angles whose sum is 180 degrees. - 180 = supplementary - Same- side interior & exterior angles |___| - Angles don't have to touch in order to be supplementary. Ex: In the picture, if the measure of angle 1 = 83 degrees and we are told to find measure of angle 2, we know that measure of angle 1 + measure of angle 2 = 180 degrees.....So all we have to do now is 180(measure of angle 1 and 2) -minus 83 (measure of angle 1) = 97(measure of angle 2) because they are supplementary.

Complementary Angles

2 angles whose sum is 90 degrees -form a right angle -complementary Ex: angle HGJ & angle JGK are complementary angle HGJ = 30 degrees angle JGK = 60 degrees **don't have to be touching**

Perpendicular Bisector

A segment of a line, segment, or ray that is PERPENDICULAR to the segment at its midpoint. Ex: Picture forms a perpendicular bisector

Right Angles Theorem

All right angles are congruent -Right angles (90 degrees) = |__ or __| (can be upside down of this 2 but have to be 90 degrees) Ex: Top left corner: angle 1 Top right corner: angle 2 Bottom left corner: angle 3 Bottom right corner: angle 4 -If angle 1 and 3 are right angles, then angles 1 and 3 are congruent. -If angle 2 and 4 are right angles, then angles 2 and 4 are congruent.

Vertical Angles

Angles opposite of each other on two intersecting lines **ALWAYS congruent (opposite) Ex: Angles 1 and 3 & angles 2 and 4 both lie opposite of one another of the same intersecting lines.(look at picture)

Adjacent Angles

Angles touching each other with one common side -2 coplanar angles -common side -common vertex -no common interior points **Note: angles do NOT have to be 90 or 180 degrees Ex: Angles 1 and 2 in the picture are adjacent angles.

Congruent Angles

Angles who have the same angle measure Ex: corresponding angles and alternate interior/exterior angles (F & Z in F.U.Z) Picture--angles RQP and VTS are congruent to each other.

Definition of Congruence

Congruent angles that have the same measure

Addition Property of Equality

For real numbers a, b & c, If a = b, then a + c = b + c. -add same quantity to both sides of equation Ex: x-2 = 13 +2 +2 <- add same quantity both sides x = 15

Division Property of Equality

For real numbers a,b, & c, where c ≠ 0, and if a = b, then a/c = b/c. -divide same quantity on both sides of equation. Ex: 3x= 9 3x/3 = 9/3 <- divide same quantity both sides x = 3

Multiplication Property of Equality

For real numbers, a, b, & c, if a = b, then ac = bc. -multiply same quantity to both sides of equation Ex: (1/2)x= 5 2(1/2)x= 5(2) <- multiply same quantity both sides x = 10

Congruent Complements Theorem

If 2 angles are complements of the same angle (or of congruent angles), then the two angles are congruent. -like transitive property of congruency Ex: If 1 and 2 are complements and & 2 and 3 are complements, then 1 and 3 are supplements.(look at picture for reference)

Congruent and Supplementary Right Angles Theorem

If 2 angles are congruent and supplementary, then each is a right angle. Ex: If angle 5 and angle 6 are supplements and are congruent, there is only one possible angle measure for the both of them, 90 degrees aka a right angle. (look at picture for visual)

Congruent Supplements Theorem

If 2 angles are supplements of the same angle (or of congruent angles), then the 2 angles are congruent. -like transitive property of congruency Ex: Look at picture

Transitive Property

If a = b and b = c, then a = c. Ex: If angle x = angle y and angle y = angle q, then angle x = angle q.

Symmetric Property

If a = b, then b = a. Ex: 400 = x x = 400

Substitution Property of Equality

If a = b, then b can replace a in any expression. Ex: a= 5 b + 4 =9 5 + 4 = 9

In a plane, if two lines are parallel to the same line, then they are parallel to each other.

If a || b and b || c, then a || c <-->---------------------->a <-->--------------->-->--->b <----------------->-->---->c

Subtraction Property of Equality

If a, b, & c are real numbers and a = b, then a - c = b - c. -subtract same quantity on both sides of equation Ex: x + 3 = 20 -3 -3 <- add same quantity both sides x = 17

In a plane, if two lines are perpendicular to the same line, then they are parallel to each other.

If h __|__ j and k __|__ j, then, h || k.

Linear Pair Postulate

If two angles form a linear pair, then they are supplementary.

Perpendicular Transversal Theorem

In a plane, if a line is perpendicular to one of two parallel lines, then it is also perpendicular to the other.

If two points lie in a plane, then the line containing those points lies in the plane.

Line r lies in plane P which goes through points x and y.

Through any two points there is exactly one line.

Line r passes through points x and y is the only line that passes through both points

If two lines intersect, then they intersect at exactly one point.

Lines DB and AE intersect at point C.

Linear Pairs

Pair of adjacent angles whose uncommon sides are opposite rays. - sum = 180 degrees - all linear pairs are .... adjacent angles & supplementary angles Ex: In the picture one angle = 140 degrees and the other = 40 degrees. Both combined together = 180 degrees and that is a component that makes them linear pairs

Through any three noncolinear points there is exactly one plane containing them.

Points Q, R, and S are noncolinear. Plane P is the only plane that contains them.

Vertical Angles Theorem

Vertical angles are congruent. Ex: As seen in the picture, angles 1 and 4 & angles 2 and 3 are congruent.

Reflexive Property

a = a Ex: angle B = angle B

Two-column Proof

a style of proof in which the statements are written in the left-hand column and the reasons are written in the right-hand column. Ex: Statements | Reasons

Transversal

line that intersects 2 or more coplanar lines at distinct points. Ex: (refer to picture to understand example) If there was a line that went through points o and x, that line would be the transveral

Angle bisector

line, segment, or ray that intersects vertex of the angle -divides angle in 2 congruent sides -divides angle in half

Segment bisector

point, line, segment, or ray that passes through or is the midpoint of the segment. -divides segment into 2 congruent sides. -intersects at midpoint of segment

Triangle Angle-Sum Theorem

the sum of the measures of the angles of a triangle is 180. Ex: Measure of angle 1 + measure of angle 2 + measure of angle 3 = 180 degrees.

Parallel Postulate

through a point not on a line, there is exactly one line parallel to the given line.

Perpendicular Postulate

through a point not on a line, there is exactly one line perpendicular to a given line.