Transversals and Congruent Triangles and Quadrilaterals

Theorem 3-11

A transversal perpendicular to one of two parallel lines is perpendicular to the other.

Isosceles trapezoid

A trapezoid with legs of the same length.

Isosceles triangle

A triangle with at least two sides equal.

P14

If the hypotenuse and a leg of one right triangle are equal to the hypotenuse and leg of another right triangle, then the triangles are congruent.

Theorem 4-3

If the hypotenuse and an acute angle of one right triangle are equal to the hypotenuse and an acute angle of another right triangle, then the triangles are congruent. (HA)

Theorem 4-19

If the midpoints of two sides of a triangle are connected, the segment is parallel to the third side and measures half the length of the third side

P11

If three sides of one triangle are equal to three sides of another triangle, then the triangles are congruent.

Theorem 4-1

If two angles and a not-included side of one triangle are equal to the corresponding parts of another triangle, then the triangles are congruent. (AAS)

P13

If two angles and the included side of one triangle are equal to two angles and the included side of another triangle, then the triangles are congruent.

Theorem 4-8

If two angles of a triangle are equal, then the sides opposite them are equal.

Theorem 4-10

If two angles of a triangle are not equal, then the side opposite the larger angle is the longer side.

Theorem 4-2

If two legs of one right triangle are equal to two legs of another right triangle, then the triangles are congruent. (LL)

Theorem 3-15

If two lines are cut by a transversal so alternate interior angles are equal, then the lines are parallel.

Theorem 3-16

If two lines are cut by a transversal so that alternate exterior angles are equal, then the lines are parallel.

Postulate 10

If two lines are cut by a transversal so that corresponding angles are equal, then the lines are parallel P10 is the converse of P8 Postulate states that if the corresponding angles are equal (for example, 1 = 3) then lines l and m are parallel

Postulate 8

If two parallel lines are cut by a transversal, then the corresponding angles have equal measure

P12

If two sides and the included angle of one triangle are equal to two sides and the included angle of another triangle, then the triangles are congruent.

Theorem 4-16

If two sides of a quadrilateral are equal and parallel, then the quadrilateral is a parallelogram.

Theorem 4-9

If two sides of a triangle are not equal, then the angle opposite the longer side is the larger angle.

Rectangle

A parallelogram with four right angles.

Parallelogram

A quadrilateral with both pairs of opposite sides parallel.

Trapezoid

A quadrilateral with exactly one pair of sides parallel.

Square

A rectangle with all sides equal.

Altitude of a triangle

A segment from a vertex perpendicular to the opposite side.

Median of a triangle

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

Transversal

A line that intersects two or more coplanar lines in different points.

Rhombus

A parallelogram with all sides equal

Match the reasons with the statements in the proof to prove that lines l and m are parallel, given that m2 = 122° and m3 = 58° Given: m2 = 122° and m3 = 58° Prove: l || m

1. m<2 = 122° and m<3 = 58° (Given) 2. <3 and <5 are supplementary angles (Exterior Sides in Opposite Rays) 3. m<3 + m<5 = 180° (Definition of Supplementary Angles) 4. 58° + m<5 = 180° (First Substitution) 5. m<5 = 122° (Subtraction Property of Equality) 6. m<2 = m<5 (Second Substitution) 7. l || m (If corresponding angles are equal, then lines are parallel.)

M, N, O are the midpoints of AB, BC, and AC If the perimeter of triangle ABC is 27 then the perimeter of triangle MNO is...

13.5

If l||m and m∠6 = 4x - 15 and m∠7 = x + 30, then m∠6 =

45

In rhombus ABCD, diagonal BD= 6" and diagonal AC=8".The sides of the rhombus are...

5"

An isosceles triangle has a vertex angle of 80°. A base angle measures...

50° (180-80)/2 = 50° Angles in a triangle add up to 180° Two other angles are equal as isosceles therefore 100/2

Corresponding Angles

Angles in the same place on different lines

Alternate Exterior Angles

Angles that lay outside the parallel lines and are on opposite sides of the transversal; They are congruent.

Theorem 4-24

Base angles of isosceles trapezoids are equal.

Alternate exterior angles

Exterior angles that lie on opposite sides of the transversal with different vertices

Given: l || m; 1 = 20x + 5; 2 = 24x - 1 Using theorems pertaining to transversal and parallel lines, prove that m∠1 + m∠2 = 180° Find the value of x.

Given: l || m 1 = 20x + 5 2 = 24x - 1 If l || m then m<1 || m<2: m<1 || m<2 Co-interior angles = 180 m<1 + m<2 = 180 Given: 20x + 5 + 24x - 1 = 180 Combine Like Terms: 44x + 4 = 180 Subtraction Property of Equality: 44x = 176 Divison Property of Equality: x = 4

∠1 and ∠7 are supplementary by definition. Given: s || t Prove: 1, 7 are supplementary

Given: s || t Exterior sides in opposite rays: ∠5 and ∠7 are supplementary Definition of supplementary angles: ∠5 + ∠7 = 180° If lines are ||, corresponding angles are equal: ∠1 = ∠5 Substitution: ∠1 + ∠7 = 180°

Congruent

Having the same measure

Theorem 3-13

IF transversal cuts two parallel lines, alternate exterior angles are equal.

Theorem 4-14

If a diagonal is drawn in a parallelogram, then two congruent triangles are formed.

Theorem 4-4

If a leg and an acute angle of one right triangle are equal to a leg and an acute angle of another right triangle, then the triangles are congruent. (LA)

Theorem 3-12

If a transversal cuts two parallel lines, alternate interior angles are equal.

Theorem 4-17

If both pairs of opposite sides of a quadrilateral are equal, then the quadrilateral is a parallelogram.

Postulate 8

If parallel lines are cut by a transversal, corresponding angles are equal

Theorem 4-18

If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.

Theorem 4-12

If two sides of one triangle are equal to two sides of another triangle but the included angle of the first is larger than the included angle of the second, then the third side of the first triangle is longer than the third side of the second triangle.

Theorem 4-13

If two sides of one triangle are equal to two sides of another triangle but the third side of the first triangle is longer than the third side of the second triangle, then the included angle of the first is larger than the included angle of the second.

Theorem 3-14

In a plane, if two lines are perpendicular to a third line, then they are parallel to each other.

Alternate Interior Angles

Interior angles that lie on opposite sides of the transversal

Alternate interior angles

Interior angles that lie on opposite sides of the transversal with different vertices

Which of the following statements are true of a transversal?

It is a line It can be perpendicular to other lines It intersects two or more coplanar lines It bisects line segments

Converse

Opposite Reverses the order of the "if-then" statements

Corollary 1

Opposite angles of a parallelogram are equal

Corollary 2

Opposite sides of a parallelogram are equal

Theorem 4-5

The altitude to the base of an isosceles triangle bisects the base.

Theorem 4-7

The altitude to the base of an isosceles triangle bisects the vertex angle of the triangle.

Included angle

The angle formed by two sides of a triangle. The angle is between, and formed by, the two sides.

Theorem 4-6

The base angles of isosceles triangles are equal.

Theorem 4-15

The diagonals of a parallelogram bisect each other.

Theorem 4-20

The diagonals of a rectangle are equal.

Theorem 4-21

The diagonals of a rhombus are perpendicular.

Theorem 4-22

The diagonals of a rhombus bisect the angles of the rhombus.

Theorem 4-25

The diagonals of an isosceles trapezoid are equal.

Theorem 4-23

The median of a trapezoid is parallel to the bases and its measure is half the sum of the lengths of the bases.

Median of a trapezoid

The segment connecting the midpoint of the legs.

Included side

The side of a triangle that is the common side of two angles. The side is between the two angles.

Theorem 4-11

The sum of the lengths of any two sides of a triangle is greater than the length of the third side.

Postulate 9

Through a point not on the line, one and only one line can be drawn parallel to the line

Write the converse of this statement. If two angles are both obtuse, the two angles are equal.

Two angles are equal if they are both obtuse

Write the converse of this statement. If 2 's are supplementary, then they are not equal.

Two angles are not equal if they are supplementary

Corollary 3

Two parallel lines are equidistant apart throughout.

Which of the following sets could not represent the sides of a triangle?

Using the Theorem of Pythagoras (a² + b² = c²), let's go through each answer. {2, 2, 4} Put in the values: 2² + 2² = 4² Is it correct? (solve) Nope! 2² + 2² = 8. And 4² = 16. The answer is {2, 2, 4} :)

Interior Angles

angles on the inside of parallel lines cut by a transversal

Exterior Angles

angles on the outside of parallel lines cut by a transversal

When two lines are cut by a transversal, ___________ angles are formed.

eight

Congruent triangles

triangles that have their corresponding parts congruent

Write the converse of this statement. If 3 - 2x = 13, then x = -5.

x = -5 if 3 - 2x = 13.