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Math 329 Transformation Geometry Chapter 2. Part II

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Abstract: Side, Side, Side Congruency Condition. Given a one to one correspondence among the vertices of two. triangles, if the three sides of one triangle are congruent to the. corresponding sides of ... Given two noncongruent sides in a triangle, the angle opposite ...

Euclidean Constructions
The Exterior Angle of a Triangle
Math 329
Transformation Geometry
Chapter 2. Part II
G. Llosent
Department of Mathematics
California State University, San Bernardino
Winter 2009 329
Euclidean Constructions
The Exterior Angle of a Triangle
1 Euclidean Constructions
Properties of a Kite
2 The Exterior Angle of a Triangle
Winter 2009 329
Euclidean Constructions
Properties of a Kite
The Exterior Angle of a Triangle
The following rules apply to Euclidean Constructions:
Given two points, a unique straight line can be drawn as
well as a unique segment connecting the points.
It is possible to extend any part of a segment.
A circle can be drawn given its center and radius.
Any ﬁnite number of points can be chosen on a given line,
segment or circle.
Points of intersection of two lines, two circles, or a line and
a circle can be used to construct segments, lines or circles.
Given a line and a point in the line, a segment of a given
length can be constructed on the line.
Winter 2009 329
Euclidean Constructions
Properties of a Kite
The Exterior Angle of a Triangle
Exercise
1 Given a segment AB, draw an equilateral triangle with side
AB.
2 Given a segment, construct a perpendicular bisector of a
segment.
3 Given a line and a point on the line, construct a
perpendicular to the line through the point.
Winter 2009 329
Euclidean Constructions
Properties of a Kite
The Exterior Angle of a Triangle
Deﬁnition
A kite is a quadrilateral that has two pairs of congruent adjacent
sides.
B
B
C
A C A
D
D
Winter 2009 329
Euclidean Constructions
Properties of a Kite
The Exterior Angle of a Triangle
Theorem
The diagonals of a kite connecting the vertices where the
congruent sides intersect bisects the angles at these vertices
and is the perpendicular bisector of the other diagonal.
Proof.
Winter 2009 329
Euclidean Constructions
Properties of a Kite
The Exterior Angle of a Triangle
Deﬁnition
A kite with all sides congruent is called a rhombus.
Corollary
The diagonals of a rhombus are perpendicular bisectors of
each other, and each bisects a pair of opposite angles.
Winter 2009 329
Euclidean Constructions
Properties of a Kite
The Exterior Angle of a Triangle
Exercise
1 Prove the converse of the statement in the previous
Corollary: A quadrilateral in which the diagonals are
perpendicular bisectors of each other is a rhombus.
2 State and prove a statement similar to that given in
Problem 1 for a kite.
3 Use the properties of a kite or rhombus to bisect a
segment.
4 Use the properties of a kite or rhombus to bisect an angle.
Winter 2009 329
Euclidean Constructions
Properties of a Kite
The Exterior Angle of a Triangle
Theorem
Side, Side, Side Congruency Condition
Given a one to one correspondence among the vertices of two
triangles, if the three sides of one triangle are congruent to the
corresponding sides of the second triangle, then the triangles
are congruent.
Proof.
Winter 2009 329
Euclidean Constructions
Properties of a Kite
The Exterior Angle of a Triangle
Theorem
Hypotenuse Leg Congruence Condition
If the hypothenuse and a leg of one triangle are congruent to
the hypothenuse and a leg of another right triangle, then the
triangles are congruent.
Proof.
Exercise
Winter 2009 329
Euclidean Constructions
The Exterior Angle of a Triangle
In the following ﬁgure, ∠BAC is one of the angles of the triangle
and both ∠BAD and ∠CAE are exterior angles.
Any angle of the triangle that is not adjacent to an exterior
angle is called a remote interior angle of the exterior angle.
Thus ∠B and ∠C are remote interior angles of ∠BAD.
B
A
C
D
E
Winter 2009 329
Euclidean Constructions
The Exterior Angle of a Triangle
Theorem
The Exterior Angle Theorem
An exterior angle of a triangle is greater than either of the
remote interior angles.
Proof.
Corollary
Through a point not on a line, there is a unique perpendicular to
the line.
Proof.
Winter 2009 329
Euclidean Constructions
The Exterior Angle of a Triangle
Theorem
Hypothenuse Acute Angle Congruence Condition
If the hypothenuse and an acute angle of one right angle are
congruent to the hypothenuse and an acute angle of another
right triangle, then the triangles are congruent.
Proof.
Winter 2009 329
Euclidean Constructions
The Exterior Angle of a Triangle
Theorem
A point is on the angle bisector of an angle if and only if it is
equidistant from the sides of the angle.
Proof.
Winter 2009 329
Euclidean Constructions
The Exterior Angle of a Triangle
Theorem
Given two noncongruent sides in a triangle, the angle opposite
the longer side is greater than the angle opposite the shorter
side.
Theorem
Given two noncongruent angles in a triangle, the side opposite
the greater angle is longer than the side opposite the smaller
angle.
Winter 2009 329
Euclidean Constructions
The Exterior Angle of a Triangle
Theorem
The Triangle Inequality
The sum of the lengths of any two sides of a triangle is greater
than the length of the third side.
Proof.
Winter 2009 329
Euclidean Constructions
The Exterior Angle of a Triangle
Exercise
1 Show that given three segments a, b and c such that
a + b > c it is not always possible to construct a triangle
whose sides are a, b and c.
2 Construct three segments a, b and c so that a triangle with
these segments as sides will exist. Is the existence and
therefore the construction of the triangle assured by The
Triangle Inequality Theorem? Justify your answer.
Exercise
Solve The Hiker’s Shortest Path Problem from Day 1 and prove
your solution is right!
Winter 2009 329
Euclidean Constructions
The Exterior Angle of a Triangle
Homework
3.4: 1, 2, 3, 5, 7, 9, 10, 15, 18
3.5: 1, 2, 3, 4, 5, 6, 7, 8, 11, 13, 18
Winter 2009 329
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