G.3(E)·Geometry·Logical Reasoning & ProofSupporting

Deductive Reasoning

Use deductive reasoning, including chains of conditionals, to prove statements.

Students build deductive arguments by linking conditional statements in the correct direction and distinguishing valid chains from reversed or unsupported conclusions.

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 practice45 minutes

Learn the concept

Deduction uses accepted facts

Deductive reasoning starts with given statements, definitions, and established theorems. A conclusion must follow from those facts, rather than from how many examples happen to fit.

Link the middle condition

The law of syllogism says that if $p\rightarrow q$ and $q\rightarrow r$ , then $p\rightarrow r$ . The middle statement q connects the two arrows; it does not need to appear in the final combined conditional.

Do not reverse the chain

From $p\rightarrow q$ and $q\rightarrow r$ , you cannot automatically conclude $r\rightarrow p$ . Also, $p\rightarrow r$ and $q\rightarrow r$ do not establish a chain from p to q. Shared conclusions alone do not connect the hypotheses.

Worked example

Assume these statements are true: If two angles are complementary, their measures total 90°. If two angle measures total 90°, each is less than 90°. What follows about complementary angles?

Let p mean the angles are complementary.
Let q mean their measures total 90°, and let r mean each measure is less than 90°.
The statements form p → q → r, so p → r.
The chain does not say the angles have equal measures; 30° and 60° satisfy both given statements.

If two angles are complementary, then each is less than 90°.

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

Texas public school video

Geometry: Deductive Reasoning in Quadrilaterals GM1L18

UT IPSI · Kid2Kid / Texas Education Agency

Follow property-based deductions and identify the facts that justify each conclusion.

Watch on YouTube

Try it yourself

TEKSForge practice · unscored

1. If a triangle is equilateral, then it is equiangular. If it is equiangular, then every angle measures 60°. Write the combined conditional.
Show answer and reasoning
If a triangle is equilateral, then every angle measures 60°.
Equiangular is the matching middle condition, so the law of syllogism connects the first hypothesis to the final conclusion.
2. If a figure is a square, then it has four sides. If a figure is a rectangle, then it has four sides. Does this prove every rectangle is a square?
Show answer and reasoning
No
The conditionals share a conclusion, not a connecting middle statement. A 3-by-7 rectangle also provides a counterexample.

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 16 · 1 point

The two conditional statements shown are true: If $\angle 3$ and $\angle 4$ form a linear pair, then they are supplementary. If $\angle 3$ and $\angle 4$ are supplementary, then $m\angle 3 + m\angle 4 = 180°$ . Based on these conditional statements, which statement must also be true?

If $\angle 3$ and $\angle 4$ form a linear pair, then $m\angle 3 + m\angle 4 = 180°$ .
If $\angle 3$ and $\angle 4$ form a linear pair, then $m\angle 3 = 90°$ , and $m\angle 4 = 90°$ .
If $m\angle 3 + m\angle 4 = 180°$ , then $\angle 3$ and $\angle 4$ form a linear pair.
If $\angle 3$ and $\angle 4$ are supplementary, then $\angle 3$ and $\angle 4$ form a linear pair.

Show STAAR answer and explanation

If $\angle 3$ and $\angle 4$ form a linear pair, then $m\angle 3 + m\angle 4 = 180°$ .

By the law of syllogism, linear pair implies supplementary, which implies a sum of 180°. The first hypothesis therefore implies the final conclusion.

If $\angle 3$ and $\angle 4$ form a linear pair, then $m\angle 3 = 90°$ , and $m\angle 4 = 90°$ .: A linear pair could have measures 60° and 120°; both need not be right angles.

If $m\angle 3 + m\angle 4 = 180°$ , then $\angle 3$ and $\angle 4$ form a linear pair.: This reverses the implication; angles summing to 180° need not be adjacent.

If $\angle 3$ and $\angle 4$ are supplementary, then $\angle 3$ and $\angle 4$ form a linear pair.: Supplementary angles need not share a vertex or form a linear pair.

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Start scored STAAR practice

Learning objectives

By the end of the lesson, students can:

Match the conclusion of one conditional to the hypothesis of another.
Apply the law of syllogism to state a valid combined conditional.
Explain why reversing an implication can produce an invalid argument.

Instructional sequence

1
Engage
5 min
Arrange everyday if-then cards into a chain. Ask which ending follows from the starting condition.
2
Direct instruction
10 min
Introduce the law of syllogism with p, q, and r. Contrast a chain with two statements that merely share a conclusion.
3
Guided practice
12 min
Link geometry definitions and angle facts, underlining the identical middle condition.
4
Argument audit
12 min
Students repair arguments that reverse an arrow or add an unsupported claim about equal angle measures.
5
Closure
6 min
Complete the exit ticket using both a sentence and an arrow chain.

Misconceptions to address

““If p implies q, then q must imply p.””

Teacher move: A rectangle has four right angles, but a conclusion can be shared by more than one starting condition. Reverse statements require their own justification.

““Two supplementary angles must each be 90°.””

Teacher move: Supplementary angles total 180°. Measures such as 65° and 115° show that equality is not guaranteed.

Materials

•Conditional-chain cards
•Arrow diagrams
•Geometry definition reference sheet

Teaching tips

Require an exact match at the middle link before combining statements.
Draw arrowheads prominently so the direction remains visible.
Ask students to cite the given conditional that supports every step.

Exit ticket · 3–5 min

If a figure is a square, then it is a rectangle. If a figure is a rectangle, then its diagonals are congruent. What follows?

Show expected answer

If a figure is a square, then its diagonals are congruent. The conclusion of the first conditional matches the hypothesis of the second.

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

Deduction uses accepted facts

Link the middle condition

Do not reverse the chain

Worked example

Watch the lesson

Geometry: Deductive Reasoning in Quadrilaterals GM1L18

Try it yourself

1. If a triangle is equilateral, then it is equiangular. If it is equiangular, then every angle measures 60°. Write the combined conditional.

2. If a figure is a square, then it has four sides. If a figure is a rectangle, then it has four sides. Does this prove every rectangle is a square?

Released STAAR practice

Archived released STAAR Geometry EOC · 2013 · Item 16 · 1 point

Learning objectives

Instructional sequence

Engage

Direct instruction

Guided practice

Argument audit

Closure

Misconceptions to address