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10th grade Midpoints and bisectors of segments and angles worksheet

6 printable versionsChallenge Me10 questions · answer keyFree · no account

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Midpoints and bisectors of segments and angles

Grade 10 Math Worksheet - Challenge Me

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Circle the best answer.

  1. Read this first

    Use this

    Proof card. Given: A, B, C, D and E lie on a line in that order. B is the midpoint of AC and D is the midpoint of CE. Prove: AE = 2BD. Step 1: AB = BC and CD = DE. Step 2: BD = BC + CD. Step 3: AE = AB + BC + CD + DE. Step 4: AE = 2BC + 2CD. Step 5: AE = 2BD.

  2. Ari fills in a proof, shown on the card. What is the reason for Step 5?

    • Segment addition postulate, since BD lies inside AE.
    • Definition of midpoint, since B and D are both midpoints.
    • Substitution of Step 1 into Step 2, because AB = BC.
    • Factor out 2 and use Step 2: 2BC + 2CD = 2(BC + CD) = 2BD.

  3. Use this

    Proof card. Given: A, B and F lie on a line. Ray BD bisects angle ABC. Ray BE bisects angle CBF. Prove: m∠DBE = 90°. Step 1: m∠DBC = ½ m∠ABC and m∠CBE = ½ m∠CBF. Step 2: m∠DBE = m∠DBC + m∠CBE. Step 3: m∠DBE = ½(m∠ABC + m∠CBF). Step 4: m∠DBE = ½(180°) = 90°.

  4. Bo writes the proof shown on the card. Which fact makes Step 4 work?

    • Adjacent angles are complementary, so they add to 90° and the two halves add to 45°.
    • BD and BE are both bisectors, so ∠DBC = ∠CBE.
    • ∠ABC and ∠CBF form a straight angle, so they add to 180°.
    • Two bisectors of adjacent angles are perpendicular to each other, whatever the angles are.

  5. Esme says: if m∠ABD = m∠DBC, then ray BD bisects ∠ABC. Which example shows the claim is false?

    • ∠ABC = 60°, and BD lies inside it, making ∠ABD = ∠DBC = 30° on both sides.
    • ∠ABC = 120°, and BD splits it into two 60° angles that both lie inside ∠ABC.
    • ∠ABC = 90°, and BD lies inside it, making ∠ABD = ∠DBC = 45° on both sides.
    • ∠ABC = 120°, and BD points away from the angle, so ∠ABD = ∠DBC = 120°.

  6. M is the midpoint of AB on Jun's ribbon. AM = (x² − 4x) cm and MB = (3x − 6) cm, with positive lengths. N is the midpoint of MB. What is AN?

    • 6 cm
    • 24 cm
    • 18 cm
    • 4.5 cm

  7. Ray BD bisects ∠ABC in Kira's drawing. m∠ABD = (x² − 20)° and m∠DBC = (3x + 8)°, with positive angles. What is m∠ABC?

    • 29°
    • 58°
    • 116°
    • 8°

  8. Ray BD bisects ∠ABC and ray BE bisects ∠DBC in Luz's drawing. m∠ABC = (14x − 12)° and m∠EBC = (2x + 3)°. What is m∠ABE?

    • 22°
    • 11°
    • 44°
    • 33°

  9. A, B, C, D lie in order on a line: AB = 6, BC = 4, CD = 12. M and N are the midpoints of AB and CD. What is MN for Milo?

    • 9
    • 22
    • 13
    • 11

  10. Noah says: if m∠ABD + m∠DBC = m∠ABC, then ray BD bisects ∠ABC. Which example shows the claim is false?

    • ∠ABD = 25°, ∠DBC = 25° and ∠ABC = 50°.
    • ∠ABD = 45°, ∠DBC = 45° and ∠ABC = 90°.
    • ∠ABD = 40°, ∠DBC = 40° and ∠ABC = 80°.
    • ∠ABD = 30°, ∠DBC = 50° and ∠ABC = 80°.

  11. Pia studies the triangle on the card. Which statement is justified?

    • BD bisects AC, but nothing shows it is perpendicular to AC.
    • BD ⊥ AC just when ∠ABC = 90°, since that is the right-angle case.
    • BD ⊥ AC, because an angle bisector is perpendicular to the opposite side of any triangle.
    • BD is also perpendicular to AC and bisects AC, because triangles ABD and CBD are congruent by SAS.

  12. Towns A and B are 12 km apart on a straight road. Tala places a bus stop equally far from A and B, 8 km off the road. How far is it from A?

    • 8 km
    • 14 km
    • 6 km
    • 10 km
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10th grade Midpoints and bisectors of segments and angles worksheet – printable
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About this worksheet

Midpoints and bisectors. Every version has 10 questions, and the answer key prints on the second page.

What your child will learn

  • I can use a midpoint or a bisector to set up an equation (linear or quadratic), reject a root that makes a length or angle negative, and find the length or angle asked for.
  • I can chain midpoints and bisectors, and use bisectors of adjacent and vertical angles.
  • I can pick the reason that justifies a step, give a counterexample to a converse, and use a perpendicular bisector in a fold, a bus-stop and a chord problem.

Standard: G-CO.A.1

Words to know

midpoint: The point that cuts a segment into two equal halves.

bisect: Cut into two equal parts.

angle bisector: A ray from the corner that cuts an angle into two equal angles.

m∠DFG: The measure of angle DFG in degrees; F, the middle letter, is the corner.

tick marks, arcs: Marks on a sketch: the same mark means the same length or angle.

Common mistakes

Your child gave one half where the whole, or a different piece, was asked. Re-read which piece the question names.

Your child added the endpoints but did not divide by 2, or added the wrong two numbers. A midpoint is the average of the ends.

Your child put the answer on the wrong side of a point. Check which way along the line the answer lies.

Questions parents ask

What is the most common mistake with midpoints and bisectors of segments and angles?

Your child gave one half where the whole, or a different piece, was asked. Re-read which piece the question names.

Are these 10th grade math worksheets free to print?

Yes. All 6 printable versions at each level are free to print, with the answer key, and no account is needed.

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