Download G.5: Use informal arguments to establish facts about the angle sum

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Pomona Unified Math News
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Domain: 8 Grade Geometry (G)
G.5: Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the
angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of
triangles. For example, arrange three copies of the same triangle so that the sum of the three angles appears to
form a line, and give an argument in terms of transversals why this is so.
Attributes of an Angle
1
Suggested Standards for Mathematical
• Vertex: The vertex is the common endpoint of
Practice (MP):
the rays that form the angle. Point B is the
MP.1 Solve problems and persevere in solving
figure above is the vertex of the angle ∠ABC.
them: As students begin to work with geometric
• Legs: The legs (sides) of an angle are the two
arguments, they encounter some difficulty in
rays that meet at the vertex to form the angle..
determining what is evidence for a proof and what
In the figure above, the line rays AB and BC
is not.
are the legs of the angle ∠ABC.
MP 2 Reason abstractly and quantitatively:
• Interior: The area between the rays that make
Students solve for unknown values of angle
up an angle, and extending away from the
measurements using angle criterion to do this they
vertex to infinity.
reason both quantitatively and abstractly.
• Exterior: All the space on the plane that is not
MP.5 Use appropriate tools strategically.
within the interior of the angle.
Students use measurement and technology tools to
Transversal: A line that cuts across two or more
create and study angle measurements.
(usually parallel) lines.
MP.6 Attend to precision.
Students are concerned with the proper naming of
angles and segments and with associating the
correct units for each. In addition, they make use
of academic vocabulary.
MP.7 and Look for and make use of structure:
Students build an understanding of the connections
between number, algebra and geometry as they
build multiple ways to engage with angles and
Corresponding angle: Corresponding angles are
parallel lines.
created where a transversal crosses other (usually
parallel) lines. The corresponding angles are the
Vocabulary:
ones at the same location at each intersection.
Parallel lines: Lines are parallel if they lie in the
same plane, and are the same distance apart over
their entire length. Parallel lines remain the same
distance apart over their entire length. No matter
how far you extend them, they will never meet. The
symbol for parallel is ||. We write 𝑃𝑄 || 𝑅𝑆 which is
read as "the line segment PQ is parallel to the
segment RS."
Similar: Two figures that have the same shape are
said to be similar. When two figures are similar, the
ratios of the lengths of their corresponding sides are
equal. The symbol for similar is ~.
Angle: A shape, formed by two lines or rays
The two quadrilaterals below are similar:
diverging from a common point (the vertex).
1
Adapted from Georgia Math Grade 8 flip book and
www.mathisfun.com
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Pomona Unified Math News
th
Domain: 8 Grade Geometry (G)
G.5: Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the
angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of
triangles. For example, arrange three copies of the same triangle so that the sum of the three angles appears to
form a line, and give an argument in terms of transversals why this is so.
Quad. LMNO ~ Quad. PQRS
∠ L ≡ ∠P
∠ M ≡ ∠Q
∠ N ≡ ∠R
∠ O ≡ ∠S
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Explanations and Examples:
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Congruent: Two objects are congruent if they have
the same dimensions and shape. Two angles are
congruent if they have the same measure.
∆ PRQ ≡ ∆ LKJ
∠ P ≡ ∠L
∠ R ≡ ∠K
∠ Q ≡ ∠J
𝑃𝑅 ≡ 𝐿𝐾
𝑅𝑄 ≡ 𝐾𝐽
𝑄𝑃 ≡ 𝐽𝐿
Alternate interior angle: Alternate Interior Angles
are created where a transversal crosses two (usually
parallel) lines. Each pair of these angles is inside
the parallel lines, and on opposite sides of the
transversal.
Alternate exterior angle: Alternate Exterior
Angles are created where a transversal crosses two
(usually parallel) lines. Each pair of these angles is
outside the parallel lines, and on opposite sides of
the transversal.
• Students use exploration and deductive
reasoning to determine relationships that exist
between:
a) Angle sums and exterior angle sums of
triangles,
b) Angles created when parallel lines are cut by
a transversal, and
c) The angle-angle criterion for similarity of
triangle.
• Students construct various triangles and find the
measures of the interior and exterior angles.
(MP 1 & 5)
• Students make conjectures about the relationship
between the measure of an exterior angle and the
other angles of a triangle. Using these
relationships, students use deductive reasoning
to find the measure of missing angles. (MP 3)
• Students construct parallel lines and a
transversal to examine the relationships between
the created angles. (MP 1,2 3)
• Students recognize vertical angles, adjacent
angles and supplementary angles from 7th grade
and build on these relationships to identify other
pairs of congruent angles. Using these
relationships, students use deductive reasoning
to find the measure of missing angles. (MP 1, 2,
3)
• Students construct various triangles having line
segments of different lengths but with two
corresponding congruent angles. Comparing
ratios of sides will produce a constant scale
factor, meaning the triangles are similar. (M 5, 6
7)
• Students can informally prove relationships with
transversals. (3, 7)
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Pomona Unified Math News
th
Domain: 8 Grade Geometry (G)
G.5: Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the
angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of
triangles. For example, arrange three copies of the same triangle so that the sum of the three angles appears to
form a line, and give an argument in terms of transversals why this is so.
Example 1:
Example 3
Show that 𝑚∠3 + m∠4 + m∠5 = 180° if l and m are
parallel lines and t1 & t2 are transversals.
Right triangle ABC and right triangle ACD overlap
as shown below. Angle DAC measures 20º and
angle BCA measures 30º.
Sample Response:
m∠1 + m∠2 + m∠3 = 180°. Angle 1 and Angle 5 are
congruent because they are corresponding angles
formed by a transversal across parallel lines (∠5 ≅
∠1). ∠1 can be substituted for ∠5 so now we have
m∠5 + m∠2 + m∠3 = 180°.
∠4 ≅ ∠2: because alternate interior angles are
congruent as they are formed by a transversal
across parallel line. ∠4 can be substituted for ∠2.
We now have the following:
m∠5 + m∠4 + m∠3 = 180°.
We can reorder these to say,
m∠3 + m∠4 + m∠5 = 180°
•
Students can informally conclude that the sum of
a triangle is 180º (the angle-sum theorem) by
applying their understanding of lines and
alternate interior angles.
Example 2:
In the figure below, line WX is parallel to line YZ:
X
W
a°
35°
Y
b°
c°
WX || YZ
80°
What are the values of x and y?
Possible Solution:
x = 40 and y = 40
Δ ABC :
∠ BAC = x+20°, ∠B = 90°, ∠BCA =30°
∠ BAC + ∠B + ∠ACD = 180°
x+20° + 90° + 30° = 180°
x+ 140° = 180°
-140° -140°
x = 40°
Δ ACD :
∠ CAD= 20°, ∠𝐴CD = 30°+ y ∠D = 90°
∠ CAD + ∠ACD + ∠D = 180°
20 + 30 +y + 90 = 180
140 + y = 180
-140
-140
y = 40
*Students may incorrectly assume that x + 20 must
equal y + 30.
Z
What is the measure of the angle labeled b?
Possible Solution:
• Because line WX is parallel to line YZ, and
Angle y measures 35°, then Angle a measures
35° because it is alternate interior with Angle y.
• Angle c is 80° because it is alternate interior to
Angle Z that measures 80°.
• Since the three angles form a line, the sum of
their measures is 180°, therefore Angle b must
be 65°
Web Help Links: (Use a QR scanner to take you
directly to the website)
https://learnzillion.com/resources/6877
https://learnzillion.com/resources/7033
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Pomona Unified Math News
th
Domain: 8 Grade Geometry (G)
G.5: Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the
angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of
triangles. For example, arrange three copies of the same triangle so that the sum of the three angles appears to
form a line, and give an argument in terms of transversals why this is so.
https://learnzillion.com/resources/6928
https://learnzillion.com/resources/6931
https://learnzillion.com/resources/6956