Chapter 3 Parallel Lines and Planes

Same side interior

< 1 and < 8 are what type of angles?

Corresponding

< 3 and < 5 are what type of angles?

Alternate interior

< 4 and < 5 are what type of angles?

Non-adjacent angles formed by a transversal; on opposite sides of the transversal and BETWEEN the lines

Alternate Interior angles

the lines are parallel

Complete the following statement: "If two lines are cut by a transversal and alternate interior angles are congruent, then..."

the lines are parallel

Complete the following statement: "If two lines are cut by a transversal and corresponding angles are congruent, then..."

the lines are parallel

Complete the following statement: "If two lines are cut by a transversal and same-side interior angles are supplementary then..."

Corresponding angles are congruent, alternate interior angles are congruent, and same-side interior angles are supplementary.

Complete the following statement: "If two parallel lines are cut by a transversal then..."

Non-adjacent angles formed by a transversal; One is interior, one is exterior, both are on the same side of the transversal

Corresponding Angles

A line segment joining any two non-consecutive vertices

Define a diagonal of a polygon

An angle formed by extending a line of a triangle

Define an exterior angle of a triangle

360/n

Finish the following sentence... "The measure of each exterior angle, one at each side, of any regular polygon with n-sides is..."

[180(n-2)]/n

Finish the following sentence... "The measure of each interior angle of any regular polygon with n-sides is..."

360

Finish the following sentence... "The sum of the exterior angles, one at each vertex, of any convex polygon is..."

180(n-2)

Finish the following sentence... "The sum of the interior angles of any convex polygon with n-sides is..."

A figure formed by coplanar segments (called sides) such that: (1)Each segment intersects exactly two other segments, one at each endpoint. (2)No two segments with a common endpoint are collinear.

Give the definition of a polygon

Plane AEFB, Plane CGHD and Plane AEGC, Plane BFHD

Identify two sets of parallel planes shown in the diagram

EF, CD and AC, FH

Identify two sets of skew lines in the diagram

No, it's segments are not coplanar

Is the figure a polygon? If so, is it convex or non-convex (concave)? If not, what part of the definition of a polygon does it violate?

No, not a polygon. The figure is not formed by segments

Is the figure a polygon? If so, is it convex or non-convex (concave)? If not, what part of the definition of a polygon does it violate?

No; Each segment DOES NOT intersect exactly two other segments, one at each endpoint.

Is the figure a polygon? If so, is it convex or non-convex (concave)? If not, what part of the definition of a polygon does it violate?

No; Some segments intersect more than two other segments, one at each endpoint.

Is the figure a polygon? If so, is it convex or non-convex (concave)? If not, what part of the definition of a polygon does it violate?

Yes; convex

Is the figure a polygon? If so, is it convex or non-convex (concave)? If not, what part of the definition of a polygon does it violate?

Yes; non-convex

Is the figure a polygon? If so, is it convex or non-convex (concave)? If not, what part of the definition of a polygon does it violate?

yes, convex

Is the figure a polygon? If so, is it convex or non-convex (concave)? If not, what part of the definition of a polygon does it violate?

yes; convex

Is the figure a polygon? If so, is it convex or non-convex (concave)? If not, what part of the definition of a polygon does it violate?

yes; non-convex

Is the figure a polygon? If so, is it convex or non-convex (concave)? If not, what part of the definition of a polygon does it violate?)?

< 3 & < 6; < 5 & < 4

Name all pairs of alternate interior angles

< 1 & < 5; < 3 & < 7; < 2 & < 6; < 4 & < 8

Name all pairs of corresponding angles

< 3 & < 5; < 4 & < 6

Name all pairs of same-side interior angles

Point A and point C

Name the consecutive vertices from point B

JN, JM, KM, KL, NL

Name the diagonals of polygon KNMLJ

< W

Name the exterior angle

point C

Name the non-consecutive vertices from point A

< A and < B

Name the remote interior angles

AB, BC and AC

Name the sides of triangle ABC

Line l

Name the transversal

line o, r, k, n, m, and l

Name the transversal(s)

By surface and by number of sides

Name two ways of classifying convex polygons

By sides and by angles

Name two ways of classifying triangles.

Lines on the same plane that never intersect

Parallel Lines

Non-adjacent angles formed by a transversal; on the same side of the transversal and between the lines

Same-Side Interior Angles

Non-coplanar lines that never intersect

Skew Lines

If two lines are cut by a transversal and alternate interior angles are congruent, then the lines are parallel.

State the theorem, definition, postulate or property that justifies the given statement," If < LKJ = < KJQ, then line LP // line QI."

If two lines are cut by a transversal and same-side interior angles are supplementary, then the lines are parallel.

State the theorem, definition, postulate or property that justifies the given statement," If < NMI + < LIJ = 180 , then line MO // line QI."

If two lines are cut by a transversal and corresponding angles are congruent, then the lines are parallel.

State the theorem, definition, postulate or property that justifies the given statement," If < ONK = < KJQ, then line OM // line QI."

If two lines are cut by a transversal and alternate interior angles are congruent, then the lines are parallel.

State the theorem, definition, postulate or property that justifies the given statement," If < TCX = < CXZ, then line TD // VZ."

If two lines are cut by a transversal and same-side interior angles are supplementary, then the lines are parallel.

State the theorem, definition, postulate or property that justifies the given statement," If < UYZ + VZY = 180 then line UB // VZ."

If two lines are cut by a transversal and same-side interior angles are supplementary, then the lines are parallel.

State the theorem, definition, postulate or property that justifies the given statement," If < a + < b = 180, then line 1 // line 2"

If two lines are cut by a transversal and corresponding angles are congruent, then the lines are parallel.

State the theorem, definition, postulate or property that justifies the given statement," If < a = < e, then line x // line y."

If two parallel lines are cut by a transversal, then corresponding angles are congruent.

State the theorem, definition, postulate or property that justifies the given statement," If C // E , then < B = < D."

If two parallel lines are cut by a transversal, then same side interior angles are supplementary.

State the theorem, definition, postulate or property that justifies the given statement," If k // l, then < B + E = 180."

If two parallel lines are cut by a transversal then same-side interior angles are supplementary.

State the theorem, definition, postulate or property that justifies the given statement," If line DT // line VZ then < DCX + < CXZ = 180."

If two parallel lines are cut by a transversal then corresponding angles are congruent.

State the theorem, definition, postulate or property that justifies the given statement," If line DT // line VZ, then < RST = < YZV."

If a transversal is perpendicular to one of two parallel lines, then it is perpendicular to the other one also.

State the theorem, definition, postulate or property that justifies the given statement," If line n // line p and line q is perpendicular to line n, then line p is perpendicular to line q also."

If two parallel lines are cut by a transversal, then corresponding angles are congruent.

State the theorem, definition, postulate or property that justifies the given statement," If m // n, then < 1 = < 2."

If two parallel lines are cut by a transversal, then alternate interior angles are congruent.

State the theorem, definition, postulate or property that justifies the given statement," If m // n, then < 5 = < 1."

If two parallel lines are cut by a tranversal then alternate interior angles are congruent.

State the theorem, definition, postulate or property that justifies the given statement," If r // s , then < 4 = < 6."

If two parallel lines are cut by a transversal then same-side interior angles are supplementary.

State the theorem, definition, postulate or property that justifies the given statement," If r // s , then < 4 and < 5 are supplementary."

Parallel

The symbol // means....?

A line that intersects 2 or more co-planar lines in different points.

Transversal Line

"The sum of the interior angles of any convex polygon with n-sides is 180(n-2)

Use the diagram to state the theorem, definition, postulate or property that justifies the given statement, "<1 +<2+<3+<4+<5= 540"

The sum of the exterior angles, one at each vertex, of any convex polygon is 360"

Use the diagram to state the theorem, definition, postulate or property that justifies the given statement, "<A +<B+<C+<D+<E+<F=360"

The sum of the angles of a triangle is 180 degrees

Use the diagram to state the theorem, definition, postulate or property that justifies the given statement," < a + < b + < c =180"

Through a point outside a line, there is exactly one line parallel to the given line.

Use the diagram to state the theorem, definition, postulate or property that justifies the given statement," Given line l and point P, line l ' is parallel to line l and is unique."

Through a point outside a line, there is exactly one line perpendicular to the given line.

Use the diagram to state the theorem, definition, postulate or property that justifies the given statement," Given line segment AB and point P, line c is perpendicular to line segment AB and is unique."

Two lines parallel to a third line are parallel

Use the diagram to state the theorem, definition, postulate or property that justifies the given statement," If line l is parallel to line k and line j is parallel to line k, then line l is parallel to line j."

If two lines are cut by a transversal and corresponding angles are congruent, then the lines are parallel.

Use the diagram to state the theorem, definition, postulate or property that justifies the given statement," h // g"

If two lines are cut by a transversal and same-side interior angles are supplementary, then the lines are parallel.

Use the diagram to state the theorem, definition, postulate or property that justifies the given statement," l // m"

If two lines are perpendicular to the same line then the lines are parallel.

Use the diagram to state the theorem, definition, postulate or property that justifies the given statement," line A is parallel to line B."

If a transversal is perpendicular to one of two parallel lines, then it is perpendicular to the other one also.

Use the diagram to state the theorem, definition, postulate or property that justifies the given statement," line l is perpendicular to line t."

If two lines are cut by a transversal and alternate interior angles are congruent, then the lines are parallel.

Use the diagram to state the theorem, definition, postulate or property that justifies the given statement," m // n"

If 2 angles of a triangle are congruent to 2 angles of another triangle, then the 3rd angles are congruent.

Use the diagram to state the theorem, definition, postulate, property or corollary justifying the given statement," < C = < F"

1. In a triangle, there is at most one obtuse or one right angle. 2. If 2 angles of a triangle are congruent to 2 angles of another triangle, then the 3rd angles are congruent. 3. In a right triangle, the acute angles are complementary. 4. If a triangle is equiangular, then each angle is 60 degrees.

What are the 4 corollaries pertaining to the angles of a triangle?

In a triangle, the exterior angle is equal to the sum of the two remote interior angles.

What does the Exterior Angle Theorem state?

A statement that is easily proven by applying a theorem

What is a corollary?

A convex polygon with equal angles and congruent sides

What is a regular polygon?

A remote interior angles is one of two angles that is inside the triangle and not adjacent to (or opposite of) the exterior angle

What is a remote interior angle of a triangle?

A line added to a diagram to help in a proof

What is an auxiliary line?

A figure formed by conneting three non collinear points

What is the definition of a triangle?

Acute triangle

What type of triangle is shown?

Equilateral triangle

What type of triangle is shown?

Isosceles Triangle

What type of triangle is shown?

Obtuse triangle

What type of triangle is shown?

Right triangle

What type of triangle is shown?

Scalene triangle

What type of triangle is shown?

When you construct parallel lines using the rhombus method you are copying one side of a rhombus three different times. In doing so, you form a 4-sided figure whose sides are all equal. This figure is called a rhombus.

Why is the method used for constructing parallel lines called the rhombus method?

Yes, as indicated by the arrows on lines a and b

Yes or No: Are these lines parallel?

Yes, if two lines are cut by a transversal and alternate interior angles are congruent, then lines are parallel

Yes or No: Are these lines parallel?

Yes, if two lines are cut by a transversal and corresponding angles are congruent, then lines are parallel

Yes or No: Are these lines parallel?

no, nothing indicates the lines are parallel

Yes or No: Are these lines parallel?