HSG.CO.C.11: Proving Theorems About Parallelograms
In plain English: HSG.CO.C.11 is the Common Core geometry standard that asks students to prove theorems about parallelograms. Students prove that opposite sides and opposite angles are congruent and that the diagonals bisect each other, prove the converses that show a quadrilateral is a parallelogram, and prove that rectangles are the parallelograms with congruent diagonals. It is usually taught in high school Geometry.
Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals.
Common Core State Standards for Mathematics · Domain: Congruence (CO) · Cluster: Prove geometric theorems Also written as HSG-CO.C.11 or G-CO.11 · Official standard
Students prove the properties of parallelograms that they have measured before. Starting from the definition (a quadrilateral with both pairs of opposite sides parallel), the alternate interior angles theorem and the triangle congruence criteria, they prove that opposite sides are congruent, opposite angles are congruent and the diagonals bisect each other. A diagonal cuts the parallelogram into two congruent triangles, and most of the proofs follow from that.
Students then prove the converses, which are the tests for deciding that a quadrilateral is a parallelogram, and prove that rectangles are exactly the parallelograms with congruent diagonals. The lesson ends with a practical use: carpenters check that a frame is rectangular by measuring its diagonals.
Learning Objectives
By the end of this lesson, students will be able to:
Prove that the opposite sides and the opposite angles of a parallelogram are congruent
Prove that the diagonals of a parallelogram bisect each other
Prove that a quadrilateral is a parallelogram if its opposite sides are congruent, its opposite angles are congruent, or its diagonals bisect each other
Prove that the diagonals of a rectangle are congruent, and that a parallelogram with congruent diagonals is a rectangle
Use the theorems to find unknown lengths and angles and to classify quadrilaterals given by coordinates
Prior Knowledge Required
Students should already be comfortable with:
Proving that alternate interior angles are congruent when a transversal crosses parallel lines HSG.CO.C.9
Triangle congruence by ASA, SAS and SSS HSG.CO.B.8
The triangle angle sum theorem HSG.CO.C.10
The distance formula 8.G.B.8 and the midpoint of a segment in the coordinate plane
Each student draws a parallelogram by tracing both edges of a ruler, turning the ruler, and tracing both edges again. The four lines form a parallelogram whose shape depends on the turn.
Warm-Up Prompt
"Measure the four sides, the four angles and the two diagonals of your parallelogram. Also measure the four pieces the diagonals cut each other into. Which measurements match? Compare with a neighbor whose parallelogram looks different."
Record conjectures on the board: opposite sides match, opposite angles match, the diagonals cut each other in half, and the diagonals are usually not equal. Ask: "Everyone drew a different parallelogram, and the same things matched. Is that enough to be sure for every parallelogram?" Measurements suggest, but only a proof covers every case.
Direct Instruction20 minutes
Definition. A parallelogram is a quadrilateral with both pairs of opposite sides parallel. Use Diagram 1 and prove the theorems in this order, since each one uses the ones before it.
Opposite sides are congruent. Draw diagonal AC. Because AB ∥ DC, ∠BAC ≅ ∠DCA, and because AD ∥ BC, ∠BCA ≅ ∠DAC (alternate interior angles). AC ≅ CA, so △ABC ≅ △CDA by ASA. So AB ≅ CD and BC ≅ DA.
Opposite angles are congruent. The same triangles give ∠B ≅ ∠D. Adding the congruent angle pairs at A and C gives m∠BAD = m∠BAC + m∠CAD = m∠DCA + m∠BCA = m∠BCD, so ∠A ≅ ∠C.
The diagonals bisect each other. Proved in Guided Practice with △ABE ≅ △CDE by ASA.
Converses. (a) If both pairs of opposite sides are congruent, draw a diagonal: SSS gives congruent triangles, so alternate interior angles are congruent and the sides are parallel. (b) If the diagonals bisect each other, the vertical angles at E and SAS give △AEB ≅ △CED and △AED ≅ △CEB, so alternate interior angles are congruent and both pairs of sides are parallel. (c) If both pairs of opposite angles are congruent, say x and y, then 2x + 2y = 360°, so x + y = 180° and consecutive angles are supplementary, which makes both pairs of opposite sides parallel.
Rectangles. A rectangle has four right angles, so its opposite angles are congruent and it is a parallelogram by converse (c). In rectangle ABCD, AB ≅ DC, ∠ABC ≅ ∠DCB (right angles) and BC ≅ CB, so △ABC ≅ △DCB by SAS and AC ≅ DB. Conversely, if parallelogram ABCD has AC ≅ DB, then △ABC ≅ △DCB by SSS, so ∠ABC ≅ ∠DCB. These consecutive angles are also supplementary, so each is 90°, and the parallelogram is a rectangle.
Opposite sides
In parallelogram ABCD, AB = 3x + 4 and DC = 5x - 10. Find x and AB.
Equation: Opposite sides are congruent: 3x + 4 = 5x - 10, so x = 7 and AB = DC = 25
Opposite and consecutive angles
In parallelogram ABCD, m∠A = (4y - 20)° and m∠C = (2y + 30)°. Find m∠A and m∠B.
Equation: Opposite angles are congruent: 4y - 20 = 2y + 30, so y = 25 and m∠A = 80°. Consecutive angles are supplementary, so m∠B = 100°
Diagonals bisect each other
The diagonals of parallelogram ABCD meet at E, with AE = 2z + 3 and EC = 4z - 7. Find AC.
Equation: AE = EC: 2z + 3 = 4z - 7, so z = 5, AE = 13 and AC = 26
Converse: proving a parallelogram
Quadrilateral WXYZ has W(-2, 1), X(3, 2), Y(5, -1) and Z(0, -2). Prove it is a parallelogram.
Equation: Midpoint of WY = (1.5, 0) = midpoint of XZ, so the diagonals bisect each other and WXYZ is a parallelogram
Rectangle diagonals
Show that RSTU with R(1, 2), S(5, 0), T(7, 4) and U(3, 6) is a rectangle, using only its diagonals (Diagram 2).
Equation: Both diagonals have midpoint (4, 3), so RSTU is a parallelogram; RT = SU = √40, so it is a rectangle
After Example 4, ask why computing slopes of all four sides would also work, and why the midpoint test is shorter. After Example 5, stress the order: first show it is a parallelogram, then use congruent diagonals. An isosceles trapezoid also has congruent diagonals but is not a rectangle.
Guided Practice15 minutes
Pairs complete the proof that the diagonals of a parallelogram bisect each other. Give the statements with blank reasons first, then discuss the completed version.
Two-column proof that the diagonals of a parallelogram bisect each other
Statement
Reason
1. ABCD is a parallelogram with diagonals AC and BD meeting at E
Given
2. AB ∥ DC
Definition of parallelogram
3. ∠BAE ≅ ∠DCE and ∠ABE ≅ ∠CDE
Alternate interior angles theorem (transversals AC and BD)
4. AB ≅ CD
Opposite sides of a parallelogram are congruent
5. △ABE ≅ △CDE
ASA (steps 3 and 4)
6. AE ≅ CE and BE ≅ DE
Corresponding parts of congruent triangles are congruent
7. E is the midpoint of AC and of BD, so the diagonals bisect each other
Definition of midpoint and of bisect
Then pairs write a paragraph proof of converse (a): if AB ≅ CD and BC ≅ DA, then ABCD is a parallelogram. Ask each pair which theorem from HSG.CO.C.9 they needed (the converse of the alternate interior angles theorem) and whether they used the definition of a parallelogram at the end.
Independent Practice15 minutes
Students work alone. (1) In parallelogram JKLM, m∠J = (6w - 4)° and m∠K = (4w + 14)°. Find both angles. (Consecutive angles are supplementary: 10w + 10 = 180, w = 17, m∠J = 98° and m∠K = 82°.) (2) Is the quadrilateral with vertices (-3, -2), (3, 0), (5, 5) and (-1, 3), in that order, a parallelogram? A rectangle? (Both diagonals have midpoint (1, 1.5), so it is a parallelogram. The diagonals are √113 and 5, so it is not a rectangle.) (3) Prove that a parallelogram with one right angle is a rectangle. (The opposite angle is also 90°, and each consecutive angle is 180° - 90° = 90°.)
Closure5-10 minutes
Exit ticket: (1) List three different ways to prove that a quadrilateral is a parallelogram. (2) A quadrilateral has congruent diagonals. Is it a rectangle? Give a counterexample or a reason. (No: an isosceles trapezoid has congruent diagonals. It must first be shown to be a parallelogram.) (3) Which pair of triangles and which criterion prove that opposite sides of a parallelogram are congruent?
Differentiation Strategies
For Struggling Students
Provide Diagram 1 with the two triangles from the diagonal shaded in different colors, and a reason bank (Definition of parallelogram, Alternate interior angles theorem, Reflexive property, ASA, SAS, SSS, CPCTC)
Have students mark every pair of parallel sides with arrows and every pair of congruent angles with arcs before writing the proof
Use fill-in-the-reason proofs before asking for full proofs
For Advanced Students
Prove that if one pair of opposite sides is both parallel and congruent, the quadrilateral is a parallelogram
Prove that the diagonals of a rhombus are perpendicular, using the parallelogram theorems and the converse of the perpendicular bisector theorem
Give a coordinate proof that the diagonals of any parallelogram with vertices (0, 0), (a, 0), (a + b, c) and (b, c) bisect each other
Assessment Guidance
What to Look For
Check that students start from the definition of a parallelogram and name the parallel sides before using alternate interior angles. Strong proofs name the triangle congruence criterion and list the three matching parts. For converses, look for a clear final step: "both pairs of opposite sides are parallel, so ABCD is a parallelogram by definition". For rectangles, watch for students who conclude "rectangle" from congruent diagonals alone, without first showing the figure is a parallelogram.
02
Classroom Activities
3 Activities
1
Always a Parallelogram?
20 minGroups of 3-4
Each group gets 8 condition cards about a quadrilateral. For each card, the group decides whether the condition always makes the quadrilateral a parallelogram (or, for card 8, a rectangle). If yes, they outline a proof. If no, they sketch a counterexample.
The 8 Cards
1. Both pairs of opposite sides are congruent. (Always: SSS, then alternate interior angles.)
2. Both pairs of opposite angles are congruent. (Always: the angle sum gives supplementary consecutive angles.)
3. The diagonals bisect each other. (Always: SAS with vertical angles.)
4. One pair of opposite sides is congruent and the other pair is parallel. (Not always: an isosceles trapezoid.)
5. The diagonals are congruent. (Not always: an isosceles trapezoid.)
6. The diagonals are perpendicular. (Not always: a kite.)
7. One pair of opposite angles is congruent. (Not always: a kite.)
8. The quadrilateral is a parallelogram and its diagonals are congruent. (Always a rectangle: SSS, then supplementary congruent angles.)
Procedure
Sort the cards into "always" and "not always" piles. The group must agree on every placement
For each "always" card, write the key triangle congruence and the theorem that finishes the proof
For each "not always" card, draw a labeled counterexample on the back
Discussion Questions
Which two cards share the same counterexample?
Card 5 fails but card 8 works. What does the extra condition in card 8 add to the proof?
2
Flexible Frame
20 minPairs
Pairs build a hinged quadrilateral from two 12 cm strips and two 20 cm strips of cardboard joined at the corners with paper fasteners, placing strips of the same length opposite each other. The frame can flex into many shapes, and students test which properties survive.
Procedure
Flex the frame into three different shapes. For each, check with a ruler that opposite sides stay parallel (the distance between them is the same at both ends)
Stretch two pieces of string along the diagonals and tape them where they cross. Measure the four pieces of string in each shape
Flex the frame until the two diagonals have the same length. Measure the angles with a protractor
Write which theorem or converse explains each observation
Discussion Questions
The side lengths never change, but the angles do. Which converse explains why every shape is still a parallelogram?
Why is the frame a rectangle at exactly the moment its diagonals are equal?
Modification for Distance Learning
Students build a parallelogram in dynamic geometry software by constructing parallel lines through two vertices, measure its diagonals, drag a vertex until the diagonals match, and record the angle measures at that moment.
3
The Carpenter's Diagonal Check
15 minPairs
Builders check that a door or window frame is rectangular by measuring the two diagonals. Pairs analyze the measurements of four wooden frames and decide which ones are rectangles, using the theorems to justify each decision.
The Frames (measurements in inches, sides in order around the frame)
For each frame, first decide whether it is a parallelogram and name the converse used
For each parallelogram, decide whether it is a rectangle and name the theorem used
For Frame A, check with the Pythagorean Theorem that a 48 by 30 rectangle has diagonals of about 56.6 inches
Challenge Variation
A builder measures only the diagonals of a frame and finds them equal. Sketch a frame with equal diagonals that is not a rectangle, and explain what else the builder must check.
03
Diagrams & Visual Aids
2 diagrams
Diagram 1: The Properties of a Parallelogram
Parallelogram ABCD with vertices A(0, 0), B(6, 0), C(8, 4) and D(2, 4), drawn to scale. Navy ticks mark the congruent opposite sides, single and double arcs mark the congruent opposite angles, and blue ticks mark the halves of each diagonal, which meet at E(4, 2).
Diagram 2: A Rectangle Is a Parallelogram with Congruent Diagonals
RSTU is drawn to scale on a grid. Its diagonals share the midpoint N(4, 3), so it is a parallelogram, and both diagonals measure √40, so it is a rectangle. The slopes of consecutive sides confirm the right angles.
Directions: Show all work. Name the theorem, converse or definition that justifies each step. In proofs, give a reason for every statement.
Part 1: Using the Theorems (Problems 1-3)
In parallelogram PQRS, PQ = 2a + 9, RS = 5a - 6 and QR = a + 12. Also m∠P = (3b + 12)° and m∠R = (5b - 28)°. (a) Find a, PQ, QR and the perimeter. (b) Find b, m∠P and m∠Q.
The diagonals of parallelogram FGHJ meet at K. FK = 3m - 2, KH = m + 10, GK = 2n + 1 and KJ = 3n - 4. Find m, n, FH and GJ, and name the theorem you used.
Quadrilateral ABCD has vertices A(-3, -1), B(2, 0), C(4, 4) and D(-1, 3). (a) Prove that ABCD is a parallelogram using its diagonals. (b) Prove it again using its opposite sides. (c) Is ABCD a rectangle? Justify with the diagonals.
Part 2: Writing Proofs (Problems 4-6)
Write a two-column proof that the opposite angles of a parallelogram are congruent. Given: parallelogram ABCD. Prove: ∠B ≅ ∠D and ∠A ≅ ∠C.
Given: quadrilateral KLMN whose diagonals KM and LN bisect each other at P. Prove: KLMN is a parallelogram.
In rectangle WXYZ, WY = 4c - 5 and XZ = 2c + 9. (a) Find c and the length of each diagonal. (b) Prove that the diagonals of any rectangle are congruent. (c) A parallelogram has diagonals of 23 cm and 23 cm. Must it be a rectangle? Explain which theorem applies.
Rubric
Criterion
Full Credit (2 pts)
Partial Credit (1 pt)
No Credit (0 pts)
Theorem Use
Correct theorem or converse named for every step
One theorem missing or used in the wrong direction
Theorems not named
Proof Structure
Given, logical steps and conclusion, with a reason for each
Minor gap in the chain of reasoning
Steps do not lead to the conclusion
Computation
Lengths, angles, midpoints and distances correct
Correct setup with an arithmetic error
Setup incorrect
Classification
Parallelogram shown before rectangle is concluded
Correct conclusion with incomplete justification
Conclusion unsupported
05
Quiz: 20 Questions
Interactive, with answers
Instructions
Choose an answer for each multiple-choice question to see whether it is right and why. For the short answers and proofs, write your answer on paper before opening the model answer. Reset quiz clears all answers.
Multiple choice: pick an option to check it. Short answer: write your answer, then reveal the model answer.
0 of 20 answered · 0 correct
Question 1 of 20 · Multiple Choice
In parallelogram KLMN, KL = 14 cm and LM = 9 cm. What is MN?
Answer: B
MN is opposite KL, and opposite sides of a parallelogram are congruent, so MN = 14 cm. Choice A gives the adjacent side LM. Choice C adds the two given sides, and choice D subtracts them.
Question 2 of 20 · Multiple Choice
In parallelogram ABCD, m∠B = 68°. What is m∠D?
Answer: D
∠B and ∠D are opposite angles, so they are congruent: m∠D = 68°. Choice A, 112°, is the measure of the consecutive angles ∠A and ∠C. Choice B treats the angles as complementary, and choice C doubles 68°.
Question 3 of 20 · Multiple Choice
In parallelogram EFGH, m∠E = (3x + 15)° and m∠G = (5x - 25)°. What is m∠F?
Answer: C
Opposite angles are congruent: 3x + 15 = 5x - 25, so x = 20 and m∠E = 75°. ∠F is consecutive to ∠E, so m∠F = 180° - 75° = 105°. Choice A is m∠E, not m∠F. Choice B is x. Choice D comes from 180° - 15°, subtracting only the constant.
Question 4 of 20 · Multiple Choice
The diagonals of parallelogram ABCD meet at E, and AE = 9. What is AC?
Answer: A
The diagonals of a parallelogram bisect each other, so E is the midpoint of AC and AC = 2(9) = 18. Choice B gives only AE. Choice C halves AE instead of doubling it. Choice D forgets the theorem.
Question 5 of 20 · Multiple Choice
The diagonals of parallelogram ABCD meet at E, with BE = 5y - 8 and ED = 3y + 6. What is BD?
Answer: B
The diagonals bisect each other, so BE = ED: 5y - 8 = 3y + 6 gives y = 7 and BE = 27. BD = 2(27) = 54. Choice A is only half of the diagonal. Choice C is y, and choice D doubles y.
Question 6 of 20 · Multiple Choice
To prove that opposite sides of parallelogram ABCD are congruent, a student draws diagonal AC. Which reason justifies ∠BAC ≅ ∠DCA?
Answer: A
AC is a transversal of the parallel lines AB and DC, and ∠BAC and ∠DCA are alternate interior angles, so they are congruent. Choice B does not apply: the angles do not share a vertex. Choice C is about the whole angles at A and C, not the parts ∠BAC and ∠DCA cut off by the diagonal. Choice D uses the triangle congruence the proof has not yet reached.
Question 7 of 20 · Multiple Choice
In the proof that opposite sides of parallelogram ABCD are congruent, △ABC ≅ △CDA uses ∠BAC ≅ ∠DCA, ∠BCA ≅ ∠DAC and the shared side AC. Which criterion is this?
Answer: D
Two angles and the side between them (AC lies between the two angle pairs) make ASA. Choice A needs all three sides, which is what we are trying to prove. Choice B needs two sides. Choice C is not a congruence criterion.
Question 8 of 20 · Multiple Choice
Which condition is NOT enough to prove that a quadrilateral is a parallelogram?
Answer: C
An isosceles trapezoid has one pair of parallel sides and a congruent pair of legs, but it is not a parallelogram, so choice C is not enough. Choices A, B and D are the three converses proved in this lesson, and each is enough.
Question 9 of 20 · Multiple Choice
Which set of vertices, listed in order, forms a parallelogram?
Answer: C
For choice C, the diagonal from (0, 0) to (5, 3) and the diagonal from (4, 0) to (1, 3) both have midpoint (2.5, 1.5), so the diagonals bisect each other. In choice A the midpoints are (2.5, 1.5) and (2, 1.5). In choice B they are (2, 1.5) and (2.5, 1.5). In choice D they are (2.5, 1.5) and (2, 1.5). A common error is to check only one pair of sides.
Question 10 of 20 · Multiple Choice
The angles of a quadrilateral, in order, measure 72°, 108°, 72° and 108°. What can you conclude?
Answer: A
The opposite angles (72° and 72°, 108° and 108°) are congruent, so by the converse of the opposite angles theorem it is a parallelogram. Choice B would need 90° angles. Choice C ignores that both pairs of opposite sides are parallel. Choice D misses the converse.
Question 11 of 20 · Multiple Choice
A rectangle has sides of 9 cm and 12 cm. Its diagonals meet at E. What is the distance from E to each vertex?
Answer: B
Each diagonal is √(9² + 12²) = 15 cm. The diagonals of a rectangle are congruent and bisect each other, so E is 7.5 cm from every vertex. Choice A is the full diagonal. Choice C averages the sides, and choice D adds them.
Question 12 of 20 · Multiple Choice
Parallelogram ABCD has diagonals AC = 2k + 9 and BD = 5k - 9. For which value of k is ABCD a rectangle?
Answer: D
A parallelogram with congruent diagonals is a rectangle, so set 2k + 9 = 5k - 9: k = 6, and both diagonals are 21. Choice B is the diagonal length, not k. Choice A comes from adding 9 and 9 without dividing by 3. Choice C forgets that most parallelograms have unequal diagonals.
Question 13 of 20 · Multiple Choice
To prove that the diagonals of rectangle ABCD are congruent, a student shows △ABC ≅ △DCB. Which parts and criterion are used?
Answer: A
Opposite sides AB and DC are congruent, the included angles at B and C are right angles, and BC is shared: SAS. Choice B assumes AC ≅ DB, the statement being proved. Choice C is not a criterion. Choice D also assumes the conclusion.
Question 14 of 20 · Multiple Choice
A scissor lift is built from straight bars. Each pair of crossing bars is joined at the midpoint of both bars. Why is each quadrilateral formed by the ends of two crossing bars a parallelogram?
Answer: B
The two bars are the diagonals of the quadrilateral, and they are joined at their midpoints, so the diagonals bisect each other. By the converse, the quadrilateral is a parallelogram. Choice A is not known and is not needed. Choice C is not given (the bars cross at many angles). Choice D is not enough to make a parallelogram.
Question 15 of 20 · Short Answer
Write a two-column proof. Given: parallelogram ABCD. Prove: AB ≅ CD and BC ≅ DA.
1. ABCD is a parallelogram (given). 2. AB ∥ DC and AD ∥ BC (definition of parallelogram). 3. Draw diagonal AC. ∠BAC ≅ ∠DCA and ∠BCA ≅ ∠DAC (alternate interior angles theorem). 4. AC ≅ CA (reflexive property). 5. △ABC ≅ △CDA (ASA). 6. AB ≅ CD and BC ≅ DA (corresponding parts of congruent triangles are congruent).
Question 16 of 20 · Short Answer
Parallelogram ABCD has diagonals that meet at E. Write a paragraph proof that E is the midpoint of both diagonals.
Because AB ∥ DC, the diagonals are transversals, so ∠BAE ≅ ∠DCE and ∠ABE ≅ ∠CDE (alternate interior angles). Opposite sides of a parallelogram are congruent, so AB ≅ CD. By ASA, △ABE ≅ △CDE, so AE ≅ CE and BE ≅ DE. E is the midpoint of AC and of BD, so the diagonals bisect each other.
Question 17 of 20 · Short Answer
Given: quadrilateral ABCD with AB ≅ CD and BC ≅ DA. Prove that ABCD is a parallelogram.
Draw diagonal AC. AC ≅ CA, so △ABC ≅ △CDA by SSS. Then ∠BAC ≅ ∠DCA, and these are alternate interior angles for lines AB and DC with transversal AC, so AB ∥ DC (converse of the alternate interior angles theorem). Likewise ∠BCA ≅ ∠DAC gives BC ∥ AD. Both pairs of opposite sides are parallel, so ABCD is a parallelogram by definition.
Question 18 of 20 · Short Answer
Quadrilateral GHIJ has vertices G(0, -1), H(6, 2), I(4, 6) and J(-2, 3). Decide whether GHIJ is a parallelogram and whether it is a rectangle, using only its diagonals.
Midpoint of GI = (2, 2.5) and midpoint of HJ = (2, 2.5), so the diagonals bisect each other and GHIJ is a parallelogram. GI = √(4² + 7²) = √65 and HJ = √(8² + 1²) = √65. A parallelogram with congruent diagonals is a rectangle, so GHIJ is a rectangle. (Check: slope GH = 1/2 and slope HI = -2, which are perpendicular.)
Question 19 of 20 · Short Answer
Given: quadrilateral ABCD with ∠A ≅ ∠C and ∠B ≅ ∠D. Prove that ABCD is a parallelogram.
Let m∠A = m∠C = x and m∠B = m∠D = y. The angles of a quadrilateral add to 360° (a diagonal splits it into two triangles), so 2x + 2y = 360° and x + y = 180°. ∠A and ∠B are same-side interior angles for lines AD and BC with transversal AB, and they are supplementary, so AD ∥ BC. ∠A and ∠D are same-side interior angles for lines AB and DC with transversal AD, and they are also supplementary, so AB ∥ DC. ABCD is a parallelogram by definition.
Question 20 of 20 · Short Answer
Prove that if parallelogram ABCD has congruent diagonals AC and BD, then ABCD is a rectangle. Then explain why a builder who measures equal diagonals on a frame must also check that opposite sides are equal.
AB ≅ DC (opposite sides), BC ≅ CB and AC ≅ DB (given), so △ABC ≅ △DCB by SSS and ∠ABC ≅ ∠DCB. These are consecutive angles of a parallelogram, so they are supplementary. Two congruent supplementary angles each measure 90°, and then all four angles are 90°: ABCD is a rectangle. The theorem needs a parallelogram. An isosceles trapezoid has equal diagonals but is not a rectangle, so the builder must first confirm the frame is a parallelogram, for example by checking that opposite sides are equal.
0 of 20 answered · 0 correct
06
Frequently Asked Questions
10 Questions
What does HSG.CO.C.11 mean?
HSG.CO.C.11 means students must prove the properties of parallelograms, not just use them. The standard names four: opposite sides are congruent, opposite angles are congruent, the diagonals bisect each other, and rectangles are the parallelograms with congruent diagonals. Students also prove the converses, which are the tests for a parallelogram.
Is HSG.CO.C.11 taught in Geometry or Algebra 2?
It is usually taught in high school Geometry, often in a unit on quadrilaterals after triangle congruence. It relies on the parallel line theorems of HSG.CO.C.9 and the congruence criteria of HSG.CO.B.8.
How do you prove a quadrilateral is a parallelogram?
Show one of these: both pairs of opposite sides are parallel (the definition), both pairs of opposite sides are congruent, both pairs of opposite angles are congruent, or the diagonals bisect each other. With coordinates, the diagonal midpoint test is often the fastest: if both diagonals have the same midpoint, the quadrilateral is a parallelogram.
Why does drawing a diagonal help in parallelogram proofs?
Because it turns a question about a quadrilateral into a question about triangles. The diagonal is a transversal of both pairs of parallel sides, which gives two pairs of congruent alternate interior angles, and it is a shared side. So the two triangles are congruent by ASA, and their matching parts give the theorems.
Are the diagonals of a parallelogram congruent?
Not in general: only when the parallelogram is a rectangle. That is the last theorem in the standard. The diagonals of every parallelogram bisect each other, but they have equal length only in rectangles, including squares.
What is a common mistake with parallelogram proofs?
Assuming what must be proved, for example using "opposite sides are congruent" inside the proof of that same theorem. Another common mistake is concluding that a quadrilateral with congruent diagonals is a rectangle without first showing it is a parallelogram: an isosceles trapezoid is a counterexample.
What does "and conversely" mean in HSG.CO.C.11?
It points to the converse theorems. Each property of a parallelogram can be reversed into a test: a quadrilateral whose opposite sides are congruent, whose opposite angles are congruent, or whose diagonals bisect each other is a parallelogram. Many textbooks also prove the converse of the rectangle theorem, that a parallelogram with congruent diagonals is a rectangle, and this lesson proves both.
Where are parallelogram theorems used in real life?
Builders square up frames and foundations by checking that the diagonals are equal. Hinged parallelogram linkages, such as some desk lamps and scissor lifts, keep parts parallel as they move because opposite sides stay congruent.
How is this standard usually assessed?
Usually with a mix of algebra and proof. Students find unknown sides, angles or diagonal segments from expressions, decide whether a quadrilateral given by coordinates is a parallelogram or a rectangle, and write or complete two-column proofs of the theorems and their converses.
What comes after HSG.CO.C.11?
Students use coordinates to prove the same kind of statements algebraically (HSG.GPE.B.4), for example that a given quadrilateral is a rectangle. They also apply congruence and similarity to solve problems in other figures (HSG.SRT.B.5).
07
Related Standards
5 standards
These standards connect to HSG.CO.C.11: prerequisites to review first, parallel standards at the same level, and next steps that build on it.
