G.10(A)·Geometry·CongruenceSupporting

Congruence Transformations

Identify and describe congruence transformations on the coordinate plane.

Students identify rigid transformations from coordinate pairs and describe compositions of reflections and translations with a single rule, checking that the rule maps every corresponding vertex.

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

Basic rigid-motion rules

Reflection across the x-axis maps $(x,y)$ to $(x,-y)$ . Reflection across the y-axis maps it to $(-x,y)$ . A translation a units horizontally and b units vertically maps it to $(x+a,y+b)$ . These motions preserve distances and angle measures.

Combine operations in order

To reflect across the y-axis and then move down 4, first obtain $(-x,y)$ and then $(-x,y-4)$ . Write intermediate coordinates when the operations affect the same coordinate, because changing their order can change the result.

Test a rule against the correspondence

Match each original vertex with its labeled image. Substitute coordinates into the proposed rule and check all pairs. Equal displacement vectors suggest a translation; an axis reflection changes the appropriate signs. A composition of rigid motions still produces a congruent figure.

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 40 · 1 point

Coordinate grid from negative 10 to 10. Parallelogram ABCD has vertices A at negative 7, 1; B at negative 7, 6; C at negative 4, 8; and D at negative 4, 3. Parallelogram A′B′C′D′ has vertices A′ at 7, negative 2; B′ at 7, 3; C′ at 4, 5; and D′ at 4, 0. — Parallelogram ABCD and its imageOriginal visual from the released STAAR test.

Worked example

Triangle ABC has A(−4, 2), B(−1, 5), and C(−2, 1). Its image has A′(4, −1), B′(1, 2), and C′(2, −2). Find a rule and describe the motion.

Each x-coordinate changes sign: −4 → 4, −1 → 1, and −2 → 2.
Each y-coordinate decreases by 3: 2 → −1, 5 → 2, and 1 → −2.
Combine these changes as $(x,y)\rightarrow(-x,y-3)$ .
This is a reflection across the y-axis followed by a translation 3 units down.

$(x,y)\rightarrow(-x,y-3)$ ; reflect across the y-axis, then translate down 3 units.

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

Geometry - Describing Reflections and Translations Using Coordinate Notation

Five Minute Math · Aaron Daffern, Dallas ISD

Identify reflection and translation rules from coordinate changes.

Watch on YouTube

Try it yourself

TEKSForge practice · unscored

1. Reflect P(2, −5) across the x-axis, then translate 4 units left. Give the rule and P′.
Show answer and reasoning
$(x,y)\rightarrow(x-4,-y)$ ; $P'=(-2,5)$ .
The reflection negates y, and the translation subtracts 4 from x.
2. A(1, 2) maps to A′(4, 0), and B(3, 5) maps to B′(6, 3). Is the rule (x + 3, y − 2) consistent with both pairs?
Show answer and reasoning
Yes
Both points move 3 units right and 2 units down. The same displacement supports the translation rule for these pairs; any remaining vertices should also be checked.

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 40 · 1 point

Parallelogram $ABCD$ was transformed to form parallelogram $A'B'C'D'$ . Which rule describes the transformation that was used to form parallelogram $A'B'C'D'$ ?

$(x, -y)$
$(-x, y)$
$(x + 6, -y)$
$(-x, y - 3)$

Show STAAR answer and explanation

$(-x, y - 3)$

The image is reflected across the y-axis and translated down 3 units: for example A(−8, 1) maps to A′(8, −2). Thus the rule is $(x,y)\mapsto(-x,y-3)$ .

$(x, -y)$ : This reflects over the x-axis, leaving x unchanged and not producing the image to the right.

$(-x, y)$ : This reflects over the y-axis but omits the 3-unit downward translation.

$(x + 6, -y)$ : This reflects over the x-axis and shifts right 6; it does not map A to A′.

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Start scored STAAR practice

Learning objectives

By the end of the lesson, students can:

Write coordinate rules for reflections across the axes and translations.
Describe a reflection followed by a translation using one coordinate rule.
Verify congruence transformations with multiple corresponding points.

Instructional sequence

1
Engage
5 min
Show two congruent asymmetric polygons on a grid. Ask whether the same horizontal and vertical shift maps every vertex.
2
Direct instruction
12 min
Review reflection rules and translation notation. Contrast negating a coordinate with adding a constant.
3
Composition model
10 min
Reflect a polygon across the y-axis, then translate it vertically. Record intermediate points and combine the operations into one rule.
4
Rule testing
15 min
Students compare candidate rules against every corresponding vertex and describe the successful rule in words.
5
Closure
8 min
Complete the exit ticket and name the measurements preserved by the transformation.

Misconceptions to address

““Reflecting across the y-axis changes the y-coordinate.””

Teacher move: The y-coordinate stays fixed. The horizontal distance to the y-axis changes sides, so x changes sign.

““A rule that maps one vertex correctly must be correct.””

Teacher move: Several transformations can map one point to the same image. Test other corresponding vertices, especially points with different coordinates.

Materials

•Coordinate grids
•Tracing paper
•Vertex-pair tables
•Transformation-rule cards

Teaching tips

Keep a three-column table for original, reflected, and translated coordinates.
Ask whether a rule changes a sign, adds a constant, or does both.
Use asymmetric polygons so the effect of a reflection is visible.

Exit ticket · 3–5 min

A figure is reflected across the y-axis and then moved up 5 units. Write the rule and find the image of (−3, 2).

Show expected answer

The rule is $(x,y)\rightarrow(-x,y+5)$ . The point maps to $(3,7)$ . Both operations preserve lengths and angle measures.

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

Basic rigid-motion rules

Combine operations in order

Test a rule against the correspondence

Study released STAAR visuals

Archived released STAAR Geometry EOC · 2013 · Item 40 · 1 point

Worked example

Watch the lesson

Geometry - Describing Reflections and Translations Using Coordinate Notation

Try it yourself

1. Reflect P(2, −5) across the x-axis, then translate 4 units left. Give the rule and P′.

2. A(1, 2) maps to A′(4, 0), and B(3, 5) maps to B′(6, 3). Is the rule (x + 3, y − 2) consistent with both pairs?

Released STAAR practice

Archived released STAAR Geometry EOC · 2013 · Item 40 · 1 point

Learning objectives

Instructional sequence

Engage

Direct instruction

Composition model

Rule testing

Closure

Misconceptions to address