Angle Relationships, Similarity and Parallelograms

UNIT 1Angle Relationships, Similarity and

Parallelograms

UNDEFINED TERMS Point Line Plane

WHAT IS THE DEFINITION OF PERPENDICULAR LINES?

Lines that intersect to form right angles.

What is the symbol for perpendicular?

It looks like “”.

CAN YOU USE THE SEGMENT ADDITION POSTULATE, MIDPOINT OF A SEGMENT, ANGLE ADDITION POSTULATE, AND THE DEFINITION OF AN ANGLE BISECTOR?

Review your worksheets. What does a midpoint of a segment do? It makes two congruent segments. What does a bisector of an angle do? It makes two congruent angles.

WHAT DOES CONGRUENT MEAN? It means objects are the same shape

and size. Set them equal.

ANGLE RELATIONSHIPS WE NEED TO KNOW

Vertical angles Linear Pair Complementary angles Supplementary angles

WHAT DO VERTICAL ANGLES LOOK LIKE AND WHAT DO WE KNOW ABOUT THEM?

Vertical angles are congruent

…looks like a bow tie

WHAT DOES A LINEAR PAIR LOOK LIKE AND WHAT DO WE KNOW ABOUT THEM?

Linear pair angles have a sum of 180°

DEFINE COMPLEMENTARY AND SUPPLEMENTARY

Complementary angles have a sum of 90°.

Supplementary angles have a sum of 180°.

PARALLEL LINE THEOREMS If the lines are parallel, then

alternate interior angles are congruent. (Look for Z)

If the lines are parallel, then corresponding angles are

congruent. (Look for F) If the lines are parallel, then

consecutive interior angles are supplementary. (C – supp)

PROPERTIES Reflexive CAT or AB = AB⦟

Symmetric If CAT DOG⦟ ⦟ or If 40 = x + 2, then x + 2 = 40 Transitive If CAT DOG⦟ ⦟ then If CAT⦟

SIMILAR POLYGONS Polygons whose corresponding side lengths are proportional and corresponding angles are congruent.

WHAT IS SCALE FACTOR?

Ratio of the lengths of two corresponding sides

(always reduce)

THEOREM

If two polygons are similar then the ratio of their perimeters is equal to the ratios of the corresponding side lengths.

HOW DO WE PROVE TRIANGLES ARE SIMILAR?

AA~SAS~SSS~

AA~

If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar.

SAS~ If an angle of one triangle is

congruent to an angle of another triangle and the lengths of the sides including these angles are proportional, then the triangles are similar.

SSS~

If the lengths of the corresponding sides of two triangles are proportional, then the triangles are similar.

THEOREM A line is parallel to the third side of a

triangle if and only if it divides two sides of the triangle proportionally.

iff

DE ⃒⃒⃒⃒AC

THEOREM If two similar solids have a scale factor of a:b, then the corresponding areas have a ratio of a²:b², and corresponding volumes have a ratio of .

RATIO PROBLEM One pyramid has a height of 9 feet and

the other has a height of 12 feet. If the two pyramids are similar, then

What is the scale factor of the smaller to the larger?

Answer: 3:4 What is the scale factor of the area of

their bases? Answer: 9:16 The volume of the smaller is 28 meters

cubed. What is the volume of the larger?

Answer: 66.37 meters cubed

ISOSCELES TRIANGLE If a triangle has two congruent sides,

then the angles opposite those sides are congruent.

Or Base angles of an isosceles triangle are congruent.

WHAT IS A PARALLELOGRAM? It is a quadrilateral with two pair of opposite sides parallel.

WHAT ARE THE PROPERTIES OF A PARALLELOGRAM?

Opposite sides are parallel. Opposite sides are congruent. Opposite angles are congruent.

Consecutive angles are supplementary.

Diagonals bisect each other.

WHAT IS A RECTANGLE? A quadrilateral with four right angles.

What is another property of a rectangle?Answer: The diagonals are congruent.

WHAT IS A RHOMBUS? A quadrilateral with four congruent

sides.

What is a special property of a rhombus?

Diagonals are perpendicular.

WHAT IS A MEDIAN OF A TRIANGLE?

A median is a segment from a vertex of a triangle to the midpoint of the opposite side.

WHERE DO THE MEDIANS OF A TRIANGLE INTERSECT?

At a point called the centroid

THEOREM ABOUT THE CENTROID From the vertex to the centroid of the triangle is 2/3 the length of the median.

ANOTHER THEOREM The segment that joins the midpoints of

two sides of a triangle is parallel to the third side and is ½ the length of the third side.

What does x equal?3

ANOTHER THEOREM The length of the segment that joins the

midpoints of the legs of a trapezoid is ½ the length of the sum of the bases.