Angles and Lines

Definitions and Axioms on Angles About a Point

An angle is formed when two lines meet at a point.

Angle ABC has arms AB and AC. Its vertex is B.

Angles are named by their size; that is, the amount of rotation or the number of degrees between the initial arm and the terminal arm.

An acute angle is greater than 0° but less than 90°

angle ABC is acute.

A right angle is exactly 90°. Angle DEF is a right angle.

An obtuse angle is greater than 90° but less than 180°

ANGLE GHI is obtuse.

A straight angle is exactly 180°. ANGLE JKL is a straight angle or straight line.

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A reflex angle is greater than 180° but less than 360°. ANGLE MNO is a reflex angle.

A full circle measures 360°. The line OA has rotated through a full circle about the point O.

The sum of the six angles is 360°.

 Vertically Opposite Angle TheoremWhen straight lines intersect,the Vertically opposite angles are equal.

In the diagram above, angles 1 and 4, 2 and 5, 3 and 6 are pairs of Vertically opposite angles. That is, Angle 1 = Angle 4, Angle 2 = Angle 5, and Angle 3 = Angle 6.

Hint: Vertically -- look for a V -- the angles are across from each other in a V.

 Two angles that sum to 90° are complementary angles.Two angles that sum to 180° are supplementary angles.

Angle DEG and ANGLE GEF are complementary angles.

Angle FEG and ANGLE GEC are supplementary angles.

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Parallel lines are straight lines in the same plane which do not meet when produced for any finite distance in either direction.

 A transversal is a straight line which meetstwo or more other straight lines.

Theorems on Parallel Lines and their Angles

When a transversal meets a pair of parallel lines, it creates three classes of angles.

1) Co-Interior Angles: are between the parallel lines, same side of the transversal.

In the diagram, angles 3 and 6 , angles 4 and 5, are pairs of co-interior angles.

2) Alternate Angles: so called because they alternate side of transversal and side of // line.

if one angle is left of the transversal, above the parallel line, (angle 1)

the other is right of the transversal, below the other parallel line. (angle 7)

2 kinds: interior alternates and exterior alternates

Interior alternates are inside or between the parallel lines.

In the diagram, angles 3 and 5 are a pair of interior alternate angles.

Angles 4 and 6 are also interior alternate angles.

Exterior alternates are outside the parallel lines.

In the diagram, angles 1 and 7 are a pair of exterior alternate angles.

Angles 2 and 8 are also exterior alternate angles.

3) Corresponding Angles: are angles on the same side of the transversal and the same side of the parallel lines.

if one angle is left of the transversal, above the parallel line,(angle 1)

the other is left of the transversal, above the other parallel line.(angle 5)

in the diagram angles 2 and 6, 3 and 7, 1 and 5, 4 and 8 are pairs of corresponding angles.

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 Parallel Lines Theorem-1When a transversal meets parallel lines the alternate angles are equal the corresponding angles are equal the co-interior angles are supplementary.

 Parallel Lines Theorem-2 When a transversal meets straight lines If alternate angles are equal, the lines are parallel If corresponding angles are equal, the lines are parallel. If co-interior angles are supplementary, the lines are parallel

So, in our diagram:

Angle 3 = Angle 5 and Angle 4 = Angle 6, interior alternate angles.

Angle 1 = Angle 7 and Angle 2 = Angle 8, exterior alternate angles.

Angle 1 =Angle 5, Angle 2 = Angle 6, Angle 4 = Angle 8, Angle 3 = Angle 7, corresponding angles.

Angle 4 + Angle 5 = 180° and Angle 3 + Angle 6 = 180° pairs of co-interior angles.

EXAMPLE 1:

In the diagram, EF is a transversal cutting the parallel lines AB and CD,
forming the indicated angles. Use the vertically opposite angle theorem and
the parallel lines theorems to list the equal and supplementary angles.

Solution:

1) By the opposite angle theorem, Angle 1 = Angle 3, Angle 2 = Angle 4, Angle 5 = Angle 7
and Angle 6 = Angle 8.

2) By alternate angles, Angle 3 = Angle 5 and Angle 4 = Angle 6; Angle 1 = Angle 7
and Angle 2 = Angle 8

3) By corresponding angles, Angle 1 = Angle 5, Angle 2 = Angle 6, Angle 3 = Angle 7,
and Angle 4 = Angle 8.

4) By co-interior angles, Angle 3 + Angle 6 = 180° and Angle 4 + Angle 5 = 180° .

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AB // CD and FG is perpendicular to AB.
Find the measures of the 8 numbered angles.

2/ Find the sum of the measures of angle a and angle b.

3/ Find the measure of all the angles at E in the diagram.

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 corresponding angles 1 = 6 = 38º complement of Angle 1 Angle 2 = 90º – 38º = 52º Angle 3 = 90º given alternate angles Angle 4 = Angle 6 = 38º vertically opposite Angle 5 = Angle 2 = 52º Angle 6 is opposite to 38º Angle 6 = 38º KE is a straight line, Angle 7 = 180º – 38º = 142º Angle 8 is opposite to 7 Angle 8 = Angle 7 = 142º

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