Download Basic Geometry - Congruence Similar and Angle Relationships

MA 7C/7D
Arcadia Valley Career
and Technology Center
Topic: Basic Geometry
Show-Me Standards: MA2
Mathematics Embedded Credit
Last Update: January 2005
Focus: Similar/Congruent Figures;
Angle Relationships
MO Grade Level Expectations: G1A9,
G1B8, G3B8, M1A6
NCTM Standards: 8A, 8B
OBJECTIVE: Students will understand and apply basic geometry skills to similar and
congruent figures, as well as, angle relationships.
Introduction:
Two geometric figures that have the same size and shape are said to be congruent. Congruence is
one of the most important concepts in basic geometry. Congruence is not just based on what is
thought, it must be backed by information that can be proven.
Triangles are one of the most common shapes used in the construction industry. Triangular shapes
stabilize and brace other geometric figures. An important phrase related to triangles is:
“Corresponding Parts of Congruent Triangles are Congruent” (or, ‘CPCTC’). If two triangles match so
that all six corresponding parts are congruent, then the two triangles are congruent.
“Side-Side-Side” Postulate (SSS):
If three sides of one triangle are congruent to three sides of another triangle, then the triangles are
congruent.
“Side-Angle-Side” Postulate (SAS):
If two sides and the included angle of one triangle are congruent to the corresponding parts of
another triangle, then the triangles are congruent.
“Angle-Side-Angle” Postulate (ASA):
If two angles and the included side of one triangle are congruent to the corresponding parts of
another triangle, then the triangles are congruent.
“Angle-Angle-Side” Postulate (AAS):
If two angles and a non-included side of one triangle are congruent to the corresponding parts of
another triangle, then the triangles are congruent.
“Isosceles Triangle Theorem”:
If two sides of a triangle are congruent, then the angles opposite those sides are congruent. OR,
Base angles of an isosceles triangle are congruent.
 If a triangle is equilateral, then it must be equiangular.
“Hypotenuse-Leg Theorem” (HL):
If the hypotenuse and a leg of a right triangle are congruent to the hypotenuse and leg of a second
right triangle, then the two triangles are congruent.
Similar Triangles: Two triangles that have the exact same angles.
Supplementary Angles: if the sum of the measures of two angles is 180-degrees, the angles are
called supplementary.
Complementary Angles: if the sum of the measures of two angles is 90-degrees, then the angles
are called complementary.
Transversal: a line that intersects two or more coplanar lines each in a different point.
“Corresponding Angles” Postulate:
If a transversal intersects two parallel lines, then the corresponding angles are congruent.
“Alternate Interior Angles” Theorem:
If a transversal intersects two parallel lines, then the pairs of alternate interior angles are congruent.
“Alternate Exterior Angles” Theorem:
If a transversal intersects two parallel lines, then the pairs of alternate exterior angles are congruent.
Solve the Following Problems:
1. Kevin cuts and sews tents for an outdoor sporting goods company. He uses patterns to make
sure all the tents are the same. Kevin can use the same pattern for the two front panels of
the tent if ABF  ABC. Show that the panels are congruent and state the postulate you
used to determine this.
B
5 ft.
F
3 ft.
5 ft.
A
3 ft.
C
2. Are these two triangles congruent? What postulate applies to the congruency of the figures?
What is the value of ‘a’?
3. In a right triangle, one of the acute angles is 33-degrees. The other acute angle is
represented by 2x – 60-degrees. Find the value of x.
4. Prior to the start of a sailboat race, an official must certify that all sails are the same size.
Without unrigging the triangular sails from their masts, the official measures the horizontal
length of each sail and the angles at each end. Is this enough information for the official to
certify the sails congruent? Why or why not.
5. Two angles are complimentary. One angle is 5 times greater than the other. What is the
measure of each angle?
6. Two angles are supplementary. The measure of one is 15-degrees more than twice the
measure of the other. What is the measure of each angle?
7. Two parallel lines are intersected by a transversal, how many pairs of alternate interior angles
are formed?
8. What is the relationship between consecutive exterior angles?
9. Suppose a joint (J) is between two points (A and B) on a floor joist. Point M is the midpoint of
the line formed between points J and B ( JB ). AB = 12 cm and MB = 2.5 cm. What is the
length of JB ? What is the length of AJ ?
10. In each of the following figures, ‘s’ is the transversal of two parallel lines. Find the measure of
1and2 .
2
1
6x+50o
6x+80o
s
2
x+25o
1
18x-40o
s
11. Kristen is helping her younger brother build a scale model of the Washington Monument for a
class project. The top section of the monument is a pyramid with isosceles triangles for sides.
The measure of the vertex angle of each triangle is 35.2o. What measure should Kristen and
her brother cut for the base angles for each triangle?
12. Complete each statement with ‘less than 90-degrees’, ‘equal to 90-degrees’, or more than 90degrees’.
a. If an isosceles triangle is an acute triangle, then the vertex angle is _____________.
b. If an isosceles triangle is a right triangle, then the vertex angle is _______________.
c. If an isosceles triangle is an obtuse triangle, then the vertex angle is ____________.
13. Find the value of x in the following
Amphitheater.
Stage
representation of an
110o
x
x
14. For the following diagrams, which postulate(s) or theorem(s) prove the two triangles
illustrated are congruent? If there is no enough information, write ‘cannot tell’.