Download Lesson 6 Day 1

Lesson 6
Proving Theorems about Lines and Angles
Metalbro is a construction company involved with
building a new skyscraper in Dubai. The diagram
below is a rough sketch of a crane that Metalbro
workers are using to build the skyscraper. The vertical
line represents the support tower and the other line
represents the boom. For safety reasons, the boom
cannot be more than 15º beyond the horizontal in
either direction. A horizontal line forms a 90º angle
with the support tower. A straight line forms a 180º
angle.
Warm-Up
Essential Questions
• How do angle relationships work together in two pairs of intersecting,
opposite rays?
• How do angle relationships work together in a set of parallel lines
intersected by a transversal?
• How are angle relationships important in the real world?
• How do proofs apply to situations outside of mathematics?
Words To Know
• adjacent angles - angles that lie in the same plane and share a vertex
and a common side. They have no common interior points.
• Alternate exterior angles - angles that are on opposite sides of the
transversal and lie on the exterior of the two lines that the transversal
intersects
• Alternate interior angles - angles that are on opposite sides of the
transversal and lie within the interior of the two lines that the
transversal intersects
• Complementary angles- two angles whose sum is 90º
• Corresponding angles - angles in the same relative position with
respect to the transversal and the intersecting lines
Words To Know
• equidistant - the same distance from a reference point
• Exterior angles- angles that lie outside a pair of parallel lines
• Interior angles- angles that lie between a pair of parallel lines
• Linear pair - a pair of adjacent angles whose non-shared sides form a
straight angle
• Non-adjacent angles - angles that have no common vertex or
common side, or have shared interior points
• Perpendicular bisector - a line that intersects a segment at its
midpoint at a right angle
Words To Know
• Perpendicular lines- two lines that intersect at a right angle (90˚), written
as 𝐴𝐵 ⊥ 𝑃𝑄. Line segments and rays can also be perpendicular.
• plane - a flat, two-dimensional figure without depth that is determined by
three non-collinear points and extends infinitely in all directions
• Postulate – a true statement that does not require a proof
• Proof - if two sides and the included angle of one triangle are congruent to
two sides and the included angle of another triangle, then the two
triangles are congruent
• Right angle - an angle measuring 90˚
•
Words To Know
• Same-side exterior angles - angles that lie on the same side of the
transversal and are outside the lines that the transversal intersects;
sometimes called consecutive exterior angles
• Same-side interior angles - angles that lie on the same side of the
transversal and are in between the lines that the transversal intersects;
sometimes called consecutive interior angles
• Straight angle – an angle with rays in opposite directions; i.e., a straight
line; a straight angle measures 180º.
• supplementary angles two angles whose sum is 180º
• Traversal - a line that intersects a system of two or more lines
• Vertical Angles - nonadjacent angles formed by two pairs of opposite rays
Key Concepts
• A straight line has a constant slope and parallel lines have the same
slope.
• If a line intersects a set of parallel lines, then the angles in the same
relative position have the same measures.
• Angles in the same relative position with respect to the transversal
and the intersecting lines are corresponding angles.
• If the lines that the transversal intersects are parallel, then
corresponding angles are congruent.
Key Concepts
Key Concepts
• Alternate interior
angles are angles that
are on opposite sides of
the transversal and lie
on the interior of the
two lines that the
transversal intersects.
• If the two lines that the
transversal intersects
are parallel, then
alternate interior angles
are congruent.
Key Concepts
• Same-side exterior angles
are angles that lie on the
same side of the transversal
and are outside the lines
that the transversal
intersects.
• If the lines that the
transversal intersects are
parallel, then same-side
exterior angles are
supplementary.
• Same-side exterior angles
are sometimes called
consecutive exterior angles.
Key Concepts
• When the lines that the
transversal intersects
are parallel and
perpendicular to the
transversal, then all the
interior and exterior
angles are congruent
right angles.
Example 1
• Given 𝐴𝐵 ∥ 𝐷𝐸, prove
that ∆𝐴𝐵𝐶~∆𝐷𝐸𝐶.
Example 2
• Given l ∥ 𝑚, and line k
is a traversal.
Prove: ∠3 ≅ ∠6
Example 3
• In the following diagram ,
𝐴𝐵 ∥ 𝐶𝐷 and 𝐴𝐶 ∥ 𝐵𝐷. If
𝑚∠1 = 3 𝑥 + 15 ,
𝑚∠2 = 2𝑥 + 55, and
𝑚∠3 = 4𝑦 + 9,
find the measures of the
unknown angles and the
values of x and y.
Example 4
• In the following diagram ,
𝐴𝐵 ∥ 𝐶𝐷 and 𝐴𝐶 ∥ 𝐵𝐷. If
𝑚∠1 = 35°, 𝑚∠ 65°, find
𝑚∠𝐸𝑄𝐹
Example 5
• When a person looks at an
object, the light rays are
refracted or distorted as they
pass through the eye, and the
image is transmitted upside
down on the retina in the back
of the eye. The object and its
retinal image are similar. Prove
that they are in proportion using
similar triangles. A diagram is
given below. Assume that both
the person looking at the image
and the image are vertical.