Lesson 6-1 Line and Angle Relationships

Classify each angle or angle pair using all names that apply and name the missing measures.
1. 2.
3.
4.

5.

6.
/ Angles – Acute < 90
Right = 90
Obtuse > 90
Straight = 180
Special Pairs of Angles:
Vertical –
/ Opposite angles that are congruent / Adjacent– / Angles that are next to each other.
Complementary /
Supplementary
/
Add up to 90
/ /
Add up to 180
Lines:
Perpendicular –
lines that intersect
at right angles (⊥).
Parallel – lines that
never intersect or
cross (∥).
Transversal– a line that
intersects two or more
other lines.
Alternate Interior Angles – opposite sides of the transversal, inside the parallel lines; 3 6, 4 5.
Alternate Exterior Angles - opposite sides of the transversal, outside the parallel lines; 1 8, 2 7.
Corresponding Angles – In the same place but on a different set of angles;
1 5, 2 6, 3 7, 4 8.

Lesson 6-2 Triangles and Angles

Classify the following triangles by their angles and their sides.
1.
3. 4.
Find the value of x in each triangle:
5.
6.
7.
8. / Polygon – closed figure formed by 3 or more line segments.
Triangle – a polygon with 3 line segments.
Classifications by Angles:
Acute
/ Obtuse / Right
/ /
3 acute angles / 1 obtuse angle / 1 right
angle
Classifications by Sides:
Scalene
/ Isosceles / Equilateral
/ /
No congruent sides / 2 congruent sides / 1 right
angle
Sum of the angles of a triangle
always = 180°.
Find the value of x in ABC if mA = 80°, mB = 54°, and mC = x°.
80 + 54 + x = 180
134 + x = 180
-134 -134
x = 46°

Lesson 6-2A Sides That Form a Triangle

Will the following sides for a triangle?
1. 9, 16, 4
2. 11, 8, 9
3. 12, 11, 16
4. 6, 17, 11
5. 9, 12, 16 / Three sides will form a triangle if…
1) The 2 smaller numbers subtracted is less than the largest number.
2) The same two numbers added together are greater than the largest number.
x – yzx + y
Will 7, 9, and 24 form a triangle?
9 – 7 < 24 < 9 + 7
2 < 24 < 16
Yes or No?
If the answer is no, then they won’t form a triangle.
If the answer is yes, then they will.

Lesson 6-3 Special Right Triangles

Find the length of each missing side.
1.
2.
3.
4.
5.
6. /


The sides of a triangle whose angles measure 30º, 60º, 90º have a special relationship. The hypotenuse is always twice as long as the sides opposite the 30º angle. Then us Pythagorean Theorem to find the 3rd side.


A 45º - 45º right triangle is also an isosceles triangle because two angle measures are the same. Thus, the legs are always congruent.

Lesson 6-4 Classifying Quadrilaterals

Classify each quadrilateral using the name that best describes it.
1.
2.
3.
Find the value of x.
4.
5.
6.
7. / Quadrilateral – a polygon with 4 straight sides and 4 angles.
Copy concept map on pg 273.
Angles of a quadrilateral = 180º.


Lesson 6-5 Congruent Polygons

Determine whether the polygons shown are congruent. If so, name the corresponding parts and write a congruency statement.
1.
2.
In the figure .
Find each measurement.

3. mR
4. RT
5. mE / Congruent – two figures that have the same size and shape.
Corresponding Parts – parts of congruent polygons that “match.”
If 2 polygons are congruent;
1)Corresponding sides are equal.
2)Corresponding angles are equal.

Congruency Statement:
Angles: The arcs indicate that A ≅D, B≅ F, and C ≅E.
Sides: The sides measurements indicate that≅ , ≅ , and ≅.
Finding missing measures
Find JL.
corresponds to . So, ≅ .
Since MN = 2 cm, JL = 2 cm.
Find mH.
According to the congruence statement, B and H are corresponding angles. So, B≅ H. Since mB = 60º, mH = 60º.

Lesson 6-6 Symmetry

Determine whether the figures have line symmetry. If it does, draw all the lines of symmetry. If it doesn’t, write none.
Determine whether each figure has rotational symmetry. Write yes or no. If yes, name it’s angles of rotation.
1. 2.
3. 4.
5. Determine whether the sun symbol of New Mexico has rotational symmetry.
/ Line Symmetry – a figure can be folded so that one half of the figure coincides with the other half.
Reflection – when the image is the mirror image of another.
Line of Symmetry – the line. Also can be called the Line of Reflection.

B

Line of Symmetry
Rotations – spins. If a figure can be turned less than 360° about its center and still look like the original.

The degrees of rotational symmetry; 45°, 90°, 135°, 180°, 225°, 270°, and 315°.

Lesson 6-7 Reflections

1. Reflect over the y-axis.
2. Reflect over the x-axis.
3. Graph quadrilateral EFGH with vertices E(-4, 4), F(3, 3), G(4, 2), and H(-2, 1). Then graph the image of EFGH after a reflection over the x-axis and write the coordinates of its vertices.
4. Graph trapezoid ABCD with vertices A(1, 3), B(4, 0), C(3, -4), and D(1, -2). Then graph the image of ABCD after a reflection over the y-axis and write the coordinates of its vertices. / Steps in Drawing a Reflection:
Step 1 - Count the number of units between each vertex and the line of reflection.

Step 2 - To find the corresponding point for vertex A, move along the line, through vertex A, perpendicular to the line of reflection until you are 3 units from the line on the opposite side. Draw a point and label it A’. Repeat for each vertex.

Step 3 – Connect the new vertices to form quadrilateral A’B’C’D’.

Lesson 6-8 Translations

Draw the image of the figure after the indicated translation.
1. 5 units right and 4 units down.
2. 4 units right and 1 unit up.
3. Graph with vertices A(-2, 2), B(3, 4), and C(4, 1). Then graph the image of after a translation 2 units left and 4 unit down. Write the coordinates of its vertices. / When a figure is translated:
1) Every point is moved the same distance in the same direction.
2) The translated figure is congruent to the original figure and has the same orientation.
Step 1 – Move each vertex 2 units right, along the horizontal grid line, and then move up 3 units along the vertical grid line.

Step 2 – Connect the new vertices to create the new figure. Be sure to put the symbol beside each letter to show that it is a translation.

Lesson 6-9 Rotations

1. Graph with vertices A(1, 1), B(3, 4), and C(4, 1). Then graph the image of after a rotation of 180°about the origin, and write the coordinates of its vertices.
2. Copy and complete the quilt square by rotating the design 180° about the given point. What does the completed figure resemble? / Rotation – moves a figure about a central point.
Step 1 – Graph trapezoid ABCD.
Step 2 – To find the corresponding point for vertex A, draw a line segment between A and the origin. Then draw a second line segment starting at the origin that is the same length as the first segment and forms a 90° angle with the first segment. Draw a point at the end of the second segment and label it A’.
Step 3 – Repeat for vertex B’.

Step 4 – Repeat for vertices C’ and D’.

Lesson 6-9b Tessellations

Complete each “Your Turn” in each activity on page 304 and 305. / Tessellation – a tiling made up of copies of the same shape or shapes that fit together without gaps and without overlapping.
Transformations – movement of geometric figures.
Translation - moving a piece cut out, from one side of a regular shape, to the opposite side.


Rotation – moving a piece cut out, rotating around one axis (to the left or the right).


Glencoe Math App & Con (2006) – Course 3