Download Angles Formed by Parallel Lines

Chapter 2: Properties of Angles and Triangles
Lesson 1 - Exploring Parallel Lines
Review of Angles
Acute angle
0  x  90
Straight angle
180
Right angle
90
Reflex angle
180  x  360
Obtuse angle
90  x  180
Complete rotation
360
Properties of Triangles
Angles in a triangle - interior angles add up to 180
Isosceles triangle – two angles are equal and two side lengths are equal
Exterior angle of a polygon - angle formed by a side of a polygon and the extension of an adjacent side.
Transversal - a line that intersects two or more other lines at distinct points
Interior angles - any angles formed by a transversal and two parallel lines that lie
inside the parallel lines.
Exterior angles - any angles formed by a transversal and two parallel lines that lie
outside the parallel lines.
Corresponding angles - one interior angle and one exterior angle that are non-adjacent and on the same side
of a transversal.
Measure the 8 angles created by the transversal:
Measure the angles:
Example:
In each case, find the missing angles.
1)
3)
2)
If a transversal intersects a set of parallel lines:

2 interior angles, or exterior angles, on the same side of the transversal are supplementary.

Corresponding angles are equal

Opposite angles are equal for ANY two intersecting lines
Assignment: Page 72 #1a, 2-6
2.2 - Angles Formed by Parallel Lines
Alternate interior angles - two non-adjacent interior angles on two opposite sides of a transversal
Alternate exterior angles - two exterior angles formed between two lines and a transversal, on opposite sides of the
transversal.
Example #1
Show that the alternate interior angles are equal.
Example #2
Show that the alternate exterior angles are equal.
Example #3
Use reasoning to determine unknown angles.
Example #4
Use angles to prove that the lines are parallel.
Assignment: Page 78 # 1, 2, 3ace, 4, 6, 8, 10,
2.3 - Angle Properties in Triangles
Non-adjacent interior angles - the two angles of a triangle that do not have the same vertex as an exterior angle.
a and b are non-adjacent interior angles to exterior angle d.
*The sum of the interior angles of a triangle always add up to 180o.
Example #1
Determine the measures of the unknown angles in Δabc.
Example #2
Determine the relationship between an exterior angle of a triangle and its non-adjacent interior angles.
Example #3
Determine the measures of angles a, b, and c.
Example #4
a) Determine all 3 unknown angles.
b) Determine the 2 unknown angles.
Assignment: Page 90: # 5, 7, 10, 11, 13, 15, 16
2.4 - Angle Properties in Polygons
Convex polygon - a polygon in which each interior angle measures less than 180o.
Example of a convex polygon
Not a convex polygon (this is called a concave polygon):
Polygon
# of Sides
# of Triangles
Sum of Angle Measures
Triangle
Quadrilateral
Pentagon
Hexagon
Heptagon
Octagon
*The sum of the measures of interior angles of any n-sided convex polygon:
180(n - 2)
Example #1
Example #2
Determine the relationship between the sum of interior angles and exterior angles.
Example #3
a) Determine the total measure of interior angles of a 9-sided figure.
b) If this polygon is regular (all side lengths the same), what is the measure of each angle?
Example #4
Rearrange the interior angle formula for finding the number of sides.
Use this to find the number of sides of a polygon with interior angle total of 900o.
Example #5
Can a floor be tiled with regular octagons and squares if their side lengths are the same?
Example #6:
Determine the values of a, b, c, and d.
Assignment: Textbook: p.99 #1, 4, 8, 9, 10, 13, 16b
2.5 - Exploring Congruent Triangles
Congruent triangles - triangles that have exactly the same 3 sides and exactly the same three angles.
Triangles may be turned or flipped, but still congruent.
With a partner:
1. Each person draw a triangle (with a ruler)
2. Label 3 pieces of information about your triangle (eg. 2 side lengths and an angle measure, all 3 sides, all 3
angles, etc.)
3. Trade with your partner to see if they can replicate the triangle (without tracing, measuring, or touching the
original)
What 3 pieces of information are necessary to make it possible?
Two triangles are congruent if:
1. Side-Side-Side (SSS) Three pairs of corresponding sides are equal.
Ex.
2. Side-Angle-Side (SAS)
Two pairs of corresponding sides and the contained angle are equal.
Ex.
3. Angle-Side-Angle (ASA)
Two pairs of corresponding angles and the contained side are equal.
Ex.
Note:

These rules apply to triangles that are flipped or rotated

Not all combinations of sides and angles can show congruency
(ex. AAA, SSA/ASS are not ways of telling if triangles are congruent)
Assignment: Page 106, #1-4
2.6 - Proving Congruent Triangles
3 ways to prove congruency:
 
 1.
 
 2.
 
 3.
Example:
Show that ΔABC is congruent to ΔCBD.
Example:
Show that ΔABC is congruent to ΔDEC
Example:
Show that ΔABC is congruent to ΔDBC.
*When describing 2 triangles by their vertices, make sure the corresponding vertices are in the same order in both
descriptions.
Congruency can be used to prove other features of triangles, such as if it is isosceles or a right triangle.
Example:
Given:
 BAC =  EDC
AB = DE
AC = DC
Prove  BCE is isosceles
Assignment: Page 112: #1, 4, 5, 9, 10, 12, 14, 19