G.4(A)·Geometry·Geometric StructureSupporting

Choosing Representations

Select a pictorial, verbal, or symbolic representation to solve geometry problems.

Students choose and connect diagrams, verbal descriptions, and algebraic expressions to solve angle problems in isosceles trapezoids, including angles formed by extending the legs.

Lesson plan and worked examples by TEKSForge · not a TEA-issued lesson. Videos by independent educators.

Archived course: TEKS code and readiness label from TEA's 2013 Geometry key. Geometry is not one of Texas's current required STAAR EOC assessments, and this lesson does not change the Biology and Algebra I Study Plans.

Practice the released item Ask the tutor All Geometry released practice50 minutes

Learn the concept

Choose a diagram for spatial information

A sketch shows which lines are parallel and which rays form an angle. Mark the parallel bases and congruent base angles of an isosceles trapezoid. Extend lines only as described, and preserve the labels when redrawing.

Translate the diagram into relationships

Parallel bases let you transfer angles using corresponding angles. An interior angle and an adjacent exterior angle form a linear pair totaling 180°. In the triangle formed by the extended legs, all three interior angles total 180°.

Choose symbols for a general answer

If the two base angles of the small triangle each measure b degrees, its vertex angle is $180-2b$ . If a given exterior angle e is supplementary to b, substitute $b=180-e$ before simplifying. The expression depends on which angle is labeled.

Study released STAAR visuals

Use the original graphs, tables, and diagrams from released STAAR questions to practice reading labels, values, and relationships before answering the items below.

Archived released STAAR Geometry EOC · 2013 · Item 32 · 1 point

Horizontal lines n and v are crossed by lines r and t, which intersect above line n at the angle labeled 1. Line r passes through B on n and D on v; line t passes through A on v and C on n, forming isosceles trapezoid ABCD with the shorter base BC on top. An angle at D, between line v to the right of D and line r below D, is labeled 7y minus 4 degrees. — Lines forming trapezoid ABCDOriginal visual from the released STAAR test.

Worked example

An isosceles trapezoid has a longer horizontal base below a shorter base. The legs extend upward to P. The exterior angle beside the lower-right vertex, formed by extending the lower base to the right, is 126°. Find the angle at P.

Draw the trapezoid and mark the 126° angle outside the lower-right vertex.
The lower-right interior angle is 180° − 126° = 54°.
The two lower base angles are equal. By parallel bases, the small triangle above the trapezoid has two base angles of 54°.
The remaining angle at P is 180° − 54° − 54° = 72°.

72°

Watch the lesson

Texas public-school, Texas educator-created, and TEKS-aligned lessons, hosted by their creators. Each card names its source.

Texas public school video

8th Grade Math Parallel Lines Cut by a Transversal

Five Minute Math · Aaron Daffern, Dallas ISD

Use diagrams and angle relationships to translate parallel-line information into equations.

Watch on YouTube

Try it yourself

TEKSForge practice · unscored

1. In the same extended-leg layout, the lower interior base angles are 67°. Find the vertex angle above the trapezoid.
Show answer and reasoning
46°
The small triangle has two 67° base angles, so its vertex angle is 180° − 2(67°) = 46°.
2. Each base angle of a triangle is (4y + 9)°. Which is correct for the third angle: 180 − (4y + 9) or 180 − 2(4y + 9)?
Show answer and reasoning
180 − 2(4y + 9)
Both equal base angles must be subtracted from 180°. An annotated diagram makes the repeated angle visible.

Released STAAR practice

Texas Education Agency · archived 2013 released EOC items · scored with TEA's answer key; step-by-step explanations written by TEKSForge because this release has no rationales. Original graphs, tables, and diagrams are shown where the released item includes one.

Archived released STAAR Geometry EOC · 2013 · Item 32 · 1 point

Lines $r$ , $t$ , $n$ , and $v$ intersect as shown to form isosceles trapezoid $ABCD$ . Which expression represents the measure of $\angle 1$ in degrees?

$180 \div (7y - 4)$
$180 - 2(7y - 4)$
$(7y - 4)$
$180 - (7y - 4)$

Show STAAR answer and explanation

$180 - 2(7y - 4)$

The labeled angle at D is vertical to the trapezoid base angle there. Both base angles at A and D are $7y-4$ . The triangle formed by the extended legs therefore has apex angle $180-2(7y-4)$ .

$180 \div (7y - 4)$ : Triangle angles add to 180°; division by one base angle is not the angle-sum relationship.

$(7y - 4)$ : This is a base angle, not the apex angle.

$180 - (7y - 4)$ : This subtracts only one base angle; both equal base angles must be subtracted.

Ask the tutor about this question

Start scored STAAR practice

Learning objectives

By the end of the lesson, students can:

Annotate a diagram with parallel bases, equal base angles, and extensions.
Translate angle relationships into equations or expressions.
Choose a representation that makes a multistep angle relationship easier to see.

Instructional sequence

1
Engage
5 min
Give a verbal trapezoid description without a picture. Ask students what information a sketch would make easier to track.
2
Represent the givens
12 min
Draw an isosceles trapezoid with a horizontal longer base below a shorter base. Extend the legs upward until they meet and label the small triangle.
3
Direct instruction
10 min
Connect equal base angles, corresponding angles, linear pairs, and the triangle angle sum to symbolic expressions.
4
Guided practice
15 min
Move among words, annotated diagrams, and algebra. Compare an interior base angle with its exterior supplement before substituting expressions.
5
Closure
8 min
Complete the exit ticket and explain which representation prevented an angle-labeling error.

Misconceptions to address

““Every labeled angle near a vertex is the interior angle.””

Teacher move: Trace the two rays that form the labeled angle. An exterior angle can be supplementary to the interior angle.

““The top angle is 180° minus one base angle.””

Teacher move: The triangle has two base angles. If each is b, the remaining angle is 180° − 2b.

Materials

•Isosceles trapezoid diagrams with extended legs
•Colored pencils
•Tracing paper
•Diagram-to-equation organizer

Teaching tips

Shade the small triangle separately from the trapezoid.
Write an angle relationship in words before replacing angle measures with expressions.
Use a numerical check to reject expressions that would produce negative or impossible angles.

Exit ticket · 3–5 min

An isosceles trapezoid has its longer base below its shorter base. Its legs extend upward to meet at P. Each lower interior base angle is (3x + 8)°. Write the measure of the angle at P.

Show expected answer

The small triangle's base angles each measure $(3x+8)^\circ$ . Its angle at P is $180-2(3x+8)=164-6x$ degrees.

Sources & alignment

TEA 2013 Geometry released test TEA 2013 Geometry answer key and standards Texas Gateway, home of the UT IPSI Kid2Kid lessons

Alignment follows the standard TEA assigned to each 2013 released item. Video playback depends on YouTube access; written instruction remains available here.

Learn the concept

Choose a diagram for spatial information

Translate the diagram into relationships

Choose symbols for a general answer

Study released STAAR visuals

Archived released STAAR Geometry EOC · 2013 · Item 32 · 1 point

Worked example

Watch the lesson

8th Grade Math Parallel Lines Cut by a Transversal

Try it yourself

1. In the same extended-leg layout, the lower interior base angles are 67°. Find the vertex angle above the trapezoid.

2. Each base angle of a triangle is (4y + 9)°. Which is correct for the third angle: 180 − (4y + 9) or 180 − 2(4y + 9)?

Released STAAR practice

Archived released STAAR Geometry EOC · 2013 · Item 32 · 1 point

Learning objectives

Instructional sequence

Engage

Represent the givens

Direct instruction

Guided practice

Closure

Misconceptions to address