Geometry · Unit 2
Find missing lengths on a line, then use overlapping angles, chained midpoints, and nested bisectors.
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pieces on a line add
matching tick marks are equal
equal end pieces: overlapping angles match
a bisector cuts the angle in half
Missing lengths
Points on a line add. A longer segment is the sum of the pieces inside it.
Matching tick marks, or a ≅ statement, mean those pieces are equal.
Example: X, Y, and Z are collinear in that order, and YZ = 8.
XZ = XY + YZ
If XY = 3:
XZ = 3 + 8
XZ = 11
If XY ≅ YZ, then XY = 8.
XZ = 8 + 8
XZ = 16
Overlapping angles
Two rays lie inside an angle. When the two end pieces are equal, the two overlapping angles are equal.
Example: rays AC and AD lie inside ∠APB, and m∠APC = m∠DPB.
m∠APD = m∠APC + m∠CPD
m∠CPB = m∠CPD + m∠DPB
The end pieces match, and both overlapping angles share ∠CPD.
If m∠APD = 44°, then m∠CPB = 44°.
Chained midpoints
Point N is the midpoint of SU, and U is the midpoint of NT. The three small pieces are equal.
NT = 10 cm is two of those pieces.
Each piece = 102
Each piece = 5 cm
ST = 5 + 5 + 5 = 15 cm
Nested bisectors
A bisector cuts an angle into two equal angles. Cut one of those halves again, and each smallest piece is half of a half.
Example: ray QL bisects ∠PQR, ray QM bisects ∠PQL, and m∠PQM = 11°.
m∠PQM = m∠MQL = 11°
m∠PQL = 11° + 11° = 22°
m∠PQL = m∠LQR = 22°
m∠MQR = 11° + 22° = 33°
Points are collinear in the order shown. Each part uses the marked length.
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(a) If WX = 4, find WY.
(b) If XZ = 15, find YZ.
(c) If WX ≅ YZ and WX = 5, find WZ.
(d) If WX ≅ YZ and WZ = 29, find YZ.
(a) WY is WX and XY together.
WY = WX + XY
WY = 4 + 9
WY = 13
(b) XZ is XY and YZ together.
XZ = XY + YZ
YZ = XZ − XY
YZ = 15 − 9
YZ = 6
(c) The end pieces are congruent, and WX = 5.
WX = YZ = 5
WZ = WX + XY + YZ
WZ = 5 + 9 + 5
WZ = 19
(d) The end pieces are congruent, and WZ = 29.
WX = YZ
WZ = WX + XY + YZ
WZ = YZ + 9 + YZ
29 = YZ + 9 + YZ
29 = 2 · YZ + 9
29 − 9 = 2 · YZ
20 = 2 · YZ
YZ = 202
YZ = 10
Answer: (a) WY = 13 (b) YZ = 6 (c) WZ = 19 (d) YZ = 10
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(a) If PQ = 11, find PR.
(b) If QS = 13, find RS.
(c) If PQ ≅ RS and PQ = 6, find PS.
(d) If PQ ≅ RS and PS = 18, find RS.
(a) PR is PQ and QR together.
PR = PQ + QR
PR = 11 + 4
PR = 15
(b) QS is QR and RS together.
QS = QR + RS
RS = QS − QR
RS = 13 − 4
RS = 9
(c) The end pieces are congruent, and PQ = 6.
PQ = RS = 6
PS = PQ + QR + RS
PS = 6 + 4 + 6
PS = 16
(d) The end pieces are congruent, and PS = 18.
PQ = RS
PS = PQ + QR + RS
PS = RS + 4 + RS
18 = RS + 4 + RS
18 = 2 · RS + 4
18 − 4 = 2 · RS
14 = 2 · RS
RS = 142
RS = 7
Answer: (a) PR = 15 (b) RS = 9 (c) PS = 16 (d) RS = 7
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RW = 51, ST = 9, UV = 6, SV = 28, UW = 17
(a) Find TU.
(b) Find VW.
(c) Find SW.
(d) Find RS.
(a) SV is ST, TU, and UV together.
SV = ST + TU + UV
28 = 9 + TU + 6
28 = TU + 15
TU = 28 − 15
TU = 13
(b) UW is UV and VW together.
UW = UV + VW
VW = UW − UV
VW = 17 − 6
VW = 11
(c) SW is SV and VW together.
SW = SV + VW
SW = 28 + 11
SW = 39
(d) RW is RS and SW together.
RW = RS + SW
RS = RW − SW
RS = 51 − 39
RS = 12
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Answer: (a) TU = 13 (b) VW = 11 (c) SW = 39 (d) RS = 12
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HN = 48, JK = 7, LM = 4, JM = 24, LN = 12
(a) Find KL.
(b) Find MN.
(c) Find JN.
(d) Find HJ.
(a) JM is JK, KL, and LM together.
JM = JK + KL + LM
24 = 7 + KL + 4
24 = KL + 11
KL = 24 − 11
KL = 13
(b) LN is LM and MN together.
LN = LM + MN
MN = LN − LM
MN = 12 − 4
MN = 8
(c) JN is JM and MN together.
JN = JM + MN
JN = 24 + 8
JN = 32
(d) HN is HJ and JN together.
HN = HJ + JN
HJ = HN − JN
HJ = 48 − 32
HJ = 16
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Answer: (a) KL = 13 (b) MN = 8 (c) JN = 32 (d) HJ = 16
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FG = 8, DG = 30
(a) Find CD.
(b) Find DE and EF.
(c) Find CG.
(d) Find CE.
(a) CD and FG have one tick mark.
CD = FG
CD = 8
(b) DE and EF have two tick marks.
DE = EF
DG = DE + EF + FG
30 = DE + DE + 8
30 = 2 · DE + 8
30 − 8 = 2 · DE
22 = 2 · DE
DE = 222
DE = 11
EF = DE = 11
(c) CG is CD and DG together.
CG = CD + DG
CG = 8 + 30
CG = 38
(d) CE is CD and DE together.
CE = CD + DE
CE = 8 + 11
CE = 19
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Answer: (a) CD = 8 (b) DE = 11, EF = 11 (c) CG = 38 (d) CE = 19
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ST = 11, QT = 29
(a) Find PQ.
(b) Find QR and RS.
(c) Find PT.
(d) Find PR.
(a) PQ and ST have two tick marks.
PQ = ST
PQ = 11
(b) QR and RS have one tick mark.
QR = RS
QT = QR + RS + ST
29 = QR + QR + 11
29 = 2 · QR + 11
29 − 11 = 2 · QR
18 = 2 · QR
QR = 182
QR = 9
RS = QR = 9
(c) PT is PQ and QT together.
PT = PQ + QT
PT = 11 + 29
PT = 40
(d) PR is PQ and QR together.
PR = PQ + QR
PR = 11 + 9
PR = 20
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Answer: (a) PQ = 11 (b) QR = 9, RS = 9 (c) PT = 40 (d) PR = 20
The arcs and tick marks in the diagrams are given.
(a) If m∠WQS = 58°, find m∠RQY.
(b) If m∠WQR = 21° and m∠RQS = 15°, find m∠WQS and m∠RQY.
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(a) The two end pieces are equal, so the overlapping angles are equal.
m∠WQS = m∠WQR + m∠RQS
m∠RQY = m∠RQS + m∠SQY
Both overlapping angles use the same middle piece and equal end pieces.
m∠RQY = m∠WQS = 58°
(b) Add the pieces that make each overlapping angle.
m∠WQS = 21° + 15°
m∠WQS = 36°
m∠SQY = m∠WQR = 21°
m∠RQY = 15° + 21°
m∠RQY = 36°
Answer: (a) 58° (b) m∠WQS = 36°; m∠RQY = 36°
Part B. Ray BF bisects ∠ABC, and ray BE bisects ∠ABF. m∠ABE = 16°. Find m∠EBC.
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Part A. M is the midpoint of HK, and K is the midpoint of MJ, so HM, MK, and KJ are equal.
MJ is two of those pieces.
MJ = 182
Each piece = 9 cm
HJ = 9 + 9 + 9
HJ = 27 cm
Part B. BE bisects ∠ABF.
m∠ABE = m∠EBF = 16°
m∠ABF = 16° + 16°
m∠ABF = 32°
BF bisects ∠ABC.
m∠ABF = m∠FBC = 32°
m∠EBC = m∠EBF + m∠FBC
m∠EBC = 16° + 32°
m∠EBC = 48°
Answer: HJ = 27 cm; m∠EBC = 48°
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