Download An auxiliary line is a line that is added to a figure to aid in a proof. An

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Transcript
4-1 Congruence and Transformations
Name: ______________________________ Period:______________
Objective: Draw, identify, and describe transformations in the coordinate plane. Use properties of rigid motions to determine
whether figures are congruent and to prove figures are congruent.
Identify each transformation. Write the coordinates of the pre-image and image.
Consider the triangle whose vertices are A(-3, 2), B(-1, -4), and C(3, 0). Use a piece of graph paper to
perform the following transformations. Make sure to draw an x and y-axis below.
1. Plot the original triangle in pencil. Use the color blue to perform a translation 5 units to the left and 2
units up.
2. Go back to the original triangle in pencil. Use the color red to draw its reflection about the y-axis.
3. Use the color black to dilate the original triangle by a scale factor of 2.
4. Use the color ____________________ to rotate the image about the origin 180.
An isometry is a transformation that preserves length, angle measure, and area. A rigid transformation is another name for an
isometry. Which of the transformations above represent an isometry?
Determine whether the polygons with the given vertices are congruent.
A.
A(-3, 1), B(2, 3), C(1, 1)
P(-4, -2), Q(1, 0), R(0, -2)
Homework for section 4-1.
7. The polygon Q(3, 2), R(6, 5), S(6, 2) undergoes the transformation: (x, y) → (4x, 4y). Name the
coordinates of the image points.
8. Apply the transformation M to the polygon with the given vertices. Name the coordinates of the
image points. Identify and describe the transformation.
M(x, y)  (x
A(3, 10), B
y
C
4.2 Classifying Triangles
Objective: Classify triangles by their angle measures and side lengths. Use triangle classification to find
angle measures and side lengths.
Consider the triangle whose vertices are J(4, 8), K(10, 14), and L(16, 8). Find the length of each side of this
triangle using the Distance Formula and classify the triangle by the lengths of its sides.
Determine if the following triangles classifications are possible. An answer of “yes” is to be accompanied
by a drawing of that triangle.
a. Right, Isosceles
b. Scalene, Obtuse
c. Equilateral, Right
Homework: Textbook pages 227-228 problems 1-11, 18-19, 35-37
4-3 Angle Relationships in Triangles
Objective: Find the measures of interior and exterior angles of a triangle.
The measure of one of the acute angles in a right triangle is given. Find the
measure of the other acute angle.
3. 44.9 _____________
4. (90  z) _____________
5. 0.3 _____________
6. The measures of the angles of a triangle are in the ration 1: 4: 7. What are the measures of the angles?
(Hint: Let x, 4x, and 7x represent the angle measures.
12. Solve for x and then find the measure of
.
Let’s Prove the Triangle Sum Theorem!!! Yay!
(I’m sure you’re thinking let’s not in your head.)
Given:
Prove:
, line l is parallel to
through B.
Statements
Reasons
1.
1. Given
2.
2. Given
3.
3.
4.
4.
5.
5.
6.
6.
Prove the Exterior Angle Theorem
Statement
1
∆PQR is a triangle
Reason
Given
2
Triangle Sum Theorem
3
Definition of linear pair.
4
If two angles form a linear pair, they are
supplementary.
5
Definition of supplementary angles.
6
Statements 2, 5 and Substitution Property.
7
Subtraction Property.
Homework: Textbook page 236 problems 16-22, 24. For problem #24 write out the whole proof not just the blanks.
4-4 Congruent Triangles
Objective: Use properties of congruent triangles. Prove triangles congruent by using the definition of
congruence.