Chapter 4: Congruent Triangles

1
Chapter 4 Congruent Triangles
2
4-1 Congruent Figures

• Congruent- when two figures have the same size
  and shape

3
4-1 Continued

• Congruent triangles- two triangles are congruent
  if and only if their vertices can be matched up
  so that the corresponding parts (angles and
  sides) of the triangle are congruent
• Their corresponding angles are congruent because
  congruent triangles have the same shape.
• Their corresponding sides are congruent because
  congruent triangles have the same size.

4
4-1 Continued

• Congruent parts of triangles are marked alike.
• Congruent triangles must be named in the same
  order of congruency.

SUN
RAY
5
4-1 Continued

• When justifying statements by use of the
  definition of congruent triangles, use this
  wording
• Corresponding parts of congruent triangles are
  congruent, which is written
• Corr. Parts of s are .

6
4-1 Continued

• Congruent polygons- two polygons are congruent if
  and only if their vertices can be matched up so
  that their corresponding parts are congruent

ABFGH BCDEF
7
4-2 Some Ways to Prove Triangles Congruent

• Proving triangles congruent with only three
  corresponding parts.
• 1. Side Side Side Postulate (SSS)- if three sides
  of one triangle are congruent to three sides of
  another triangle, then the triangles are congruent

8
4-2 Continued

• Side Angle Side Postulate (SAS)- if two sides and
  the included angle of one triangle are congruent
  to two sides and the included angle of another
  triangle, then the triangles are congruent

9
4-2 Continued

• Angle Side Angle Postulate (ASA)- if two angles
  and the included side of one triangle are
  congruent to two angles and the included side of
  another triangle, then the triangles are
  congruent

10
Proof of ASA Postulate
Statement Reason
Given Prove
1. E is the midpoint of
E is the midpoint of

• Given
• Definition of a midpoint
• Given
• If two lines are perpendicular
• then they form congruent adjacent
• angles.
• 5. Reflexive property of congruence
• 6. SAS postulate

2.

3.
4.
5.
6.
11
4-3 Using Congruent Triangles

• Learning how to extract information on segments
  or angles once it is shown that they are
  corresponding parts of congruent triangles

12
4-3 Continued
Statement Reason
1.

• Given
• 2. Definition of a bisector of a segment
• 3. Definition of a midpoint
• 4. Vertical angles are congruent
• 5. SAS Postulate
• 6. Corresponding parts of congruent triangles are
  congruent
• 7. If two lines are cut by a transversal and
  alternate interior angles are congruent, then the
  lines are parallel.

and
Given AB and CD bisect each other at
M Prove AD BC
bisect each other at M
2. M is the midpoint of
and of
ll
3.

4.
5.
6.
7.
ll
13
4-3 Continued

• A line and a plane are perpendicular if and only
  if they intersect and the line is perpendicular
  to all lines in the plane that pass through the
  point of intersection.

14
4-3 Continued
Statement Reason
plane X
1.
1. Given 2. Definition of a line perpendicular to
a plane. 3. Definition of perpendicular lines 4.
Defintion of congruent angles 5. Given 6.
Reflexive Property 7. SAS postulate 8.
Corresponding parts of congruent angles are
congruent.
2.

Given PO plane X AO BO Prove PA
3. m
90 m
90
4.
5.

6.
7.
8.


15
4-3 Continued

• To prove two segments or two angles are
  congruent
• 1.) Identify two triangles in which the two
  segments or angles are corresponding parts.
• 2.) Prove that the triangles are congruent.
• 3.) State that the two parts are congruent,
  using this reason
• Corr. Parts of s are .

16
4-4 The Isosceles Triangle Theorems

• Legs- the congruent sides of a triangle
• Base- the non-congruent side of a triangle
• Base angles- the angles at the base of the
  triangle
• Vertex angle- the angle opposite the base of the
  isosceles triangle

Vertex angle
Leg
Leg
Base angles
Base
17
4-4 Continued

• The Isosceles Triangle Theorem- if two sides of a
  triangle are congruent, then the angles opposite
  those sides are congruent

18
4-4 Continued

• Corollary 1- an equilateral triangle is also
  equiangular
• Corollary 2- an equilateral triangle has three 60
  degree angles
• Corollary 3- The bisector of the vertex angle of
  an isosceles triangle is perpendicular to the
  base at its midpoint

19
4-4 Continued

• Theorem 4-2
• If two angles of a triangle are congruent, then
  the sides opposite those angles are congruent.
• Corollary- an equilateral triangle is also
  equilateral
• Theorem 4-2 is the converse of Theorem 4-1, and
  the corollary of Theorem 4-2 is the converse of
  Corollary 1 of Theorem 4-1.

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4-5 Other Methods of Proving Triangles Congruent

• Angle Angle Side Theorem (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

21
4-5 Continued

• Hypotenuse- the side opposite the right angle in
  a right triangle
• Legs- the other two sides of the triangle

hypotenuse
leg
leg
22
4-5 Continued

• Hypotenuse Leg Theorem- if the hypotenuse and a
  leg of one right triangle are congruent to the
  corresponding parts of another right triangle,
  then the triangles are congruent

23
4-5 Continued

• Leg-Leg Method- if two legs of one right triangle
  are congruent to the two legs of another right
  triangle, then the triangles are congruent
• Hypotenuse-Acute Angle Method- if the hypotenuse
  and an acute angle of one right triangle are
  congruent to the hypotenuse and an acute angle of
  another right triangle, then the triangles are
  congruent
• Leg-Acute Angle Method- If a leg and an acute
  angle of one right triangle are congruent of the
  corresponding parts in another right triangle,
  then the triangles are congruent.

24
4-6 Using More than One Pair of Congruent
Triangles
1.
1.
Statement Reason
Given Prove

• Given
• Reflexive property
• ASA postulate
• Corresponding parts of
• congruent angles are
• congruent.
• 5. Reflexive property
• 6. SAS postulate (1, 4, 5)
• 7. Corresponding parts of
• congruent angles are
• congruent.
• 8. If two lines form congruent
• adjacent angles, then the
• lines are perpendicular.

1.

2.

3.
4.

5.
6.
7.
8.



25
4-7 Medians, Altitudes, and Perpendicular
Bisectors

• Median- a segment from a vertex to the midpoint
  of the opposite side in a triangle

26
4-7 Continued
Altitude- the perpendicular segment from a vertex
to a line that contains the opposite side In
an acute triangle, the three altitudes are all
inside the right triangle.
27
4-7 Continued
In a right triangle, two of the altitudes are
parts of the triangle. They are the legs of the
right triangle. The third altitude is inside the
triangle.
In an obtuse triangle, two of the altitudes are
outside the triangle.
28
4-7 Continued
Perpendicular bisector- a line (or ray or
segment) that is perpendicular to the segment at
its midpoint
29
4-7 Continued

• Theorem 4-5
• If a point lies on the perpendicular bisector of
  a segment, then the point is equidistant from the
  endpoints of the segment.

30
4-7 Continued

• Theorem 4-6
• If a point is equidistant from the endpoints of
  a segment, then the point lies on the
  perpendicular bisector of the segment.
• Theorem 4-6 is the converse of Theorem 4-5.

31
4-7 Continued

• The distance from a point to a line (or plane) is
  defined to be the length of the perpendicular
  segment from the point to the line (or plane).

32
4-7 Continued

• Theorem 4-7
• If a point lies on the bisector of an angle,
  then the point is equidistant from the sides of
  the angle.

33
4-7 Continued

• Theorem 4-8
• If a point is equidistant from the sides of an
  angle, then the point lies on the bisector of the
  angle.
• Theorem 4-8 is the converse of Theorem 4-7.

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The End

• (Thank God!)