Download Target 5a (Day 1)

Geometry Name: _______________________________ Date: _____________ Hour: _____
Target 5a (Day 1)
Identify bisectors of angles and segments and use them to find segment measures.
Perpendicular Bisectors
Theorem 5.1 – Any point on the perpendicular ______________ of a segment is
equidistant from the ______________ of the segment.
Theorem 5.2 – Any point _________________ from the endpoints of a segment _______
on the perpendicular bisector of the segment.
Concurrent lines – When ___ or more lines intersect at a common point.
Point of Concurrency – The ________________ point where three or more lines meet.
Circumcenter – The point of concurrency of the perpendicular ______________ of a triangle.
Theorem 5.3 – (Circumcenter Theorem) The circumcenter of a triangle is
__________________ from the ________________ of the triangle.
Practice Problems
SY = 6 yd. Find TS.
Find CD if DX = 17 mm.
Each figure shows a triangle with its three perpendicular bisectors intersecting at point D.
LJ = 6 and DJ = 11. Find LD.
Find x if GQ = 9x - 12 and QF = 7x - 8.
Homework: 5a Day 1 Worksheet (from 5.1 in text, page 238)
Target 5a - continued (Day 2)
Identify bisectors of angles and segments and use them to find segment measures.
Angle Bisectors
Theorem 5.4 – Any point on the ____________ bisector is equidistant from the
______________ of the angle.
Theorem 5.5 – Any point equidistant from the sides of an _________ lies on the angle
_________________.
Incenter – The point of concurrency for _________ bisectors.
Theorem 5.6 – (Incenter Theorem) The _____________ of a triangle is equidistant from each
____________ of the triangle.
Practice Problems
Each figure shows a triangle with its three angle bisectors intersecting at point P.
mCBD 68°. Find m1.
Find PB if CP = 3.
HP = 9 and PU = 4. Find HS.
Find x if m1 = 9x + 23 and m2 =15x - 19.
Homework: 5a Day 2 Worksheet (from 5.1 in text, page 238)
Target 5b
Identify medians and altitudes of triangles and use them to find segment measures.
Medians
Median – A segment whose ____________________ are one vertex of a triangle and the
_________________ of the side opposite the vertex.
Centroid – The point of concurrency for the _________________ of a triangle.
Theorem 5.7 – (Centroid Theorem) The centroid of a triangle is located ______________ of the
distance from a ______________ to the midpoint of the side _________________
the vertex on a median.
Practice Problems
a)
is a median and ME = 4.7. Find MN.
c) Point D is a centroid.
Find x
Find y
b)
is a median. Find x.
d) Point T is a centroid. CT = 6, TJ = 20,
and KB = 33.
Find z
Find TL
Find AJ
Find TB
Altitudes
Altitude – A segment in a triangle from a ______________ to the line containing the opposite side
and _________________________________ to the line containing that side.
Orthocenter – The point of concurrency for the ____________________ of a triangle.
Practice Problems
a)
is an altitude. mXWY is 4x – 6.
Find x.
b) mTRS is 6x. Find x.
Review – Targets 5a and 5b:
Directions: Fill in the blanks with the terms for each special segment or point of concurrency
that the arrows are pointing to.
Homework: 5b Worksheet (from 5.1 in text, page 238)
Target 5c
Solve problems using triangle inequality theorems.
Quiz 5a & 5b
next time
Theorem 5.9 – If one side of a triangle is ____________ than another side, then the angle
___________ the longer side has a _________ measure than the angle opposite the shorter side.
Directions: Determine the relationship between the measures of the given angles.
a) ADB and DBA.
b) CDB and CBD.
c) RSU and SUR.
d) TSV and STV.
Theorem 5.10 – If one _______________ of a triangle has a greater measure than another angle,
then the ___________ opposite the greater angle is longer than the side opposite the lesser angle.
Directions: Determine the relationship between the lengths of the given sides.
a)
and
.
b)
and
.
c)
d)
and
and
.
.
Triangle Inequality Theorem (Theorem 5.11) – the _______ of the lengths of the two smallest
sides of a triangle is __________________ than the length of the third side.
Directions: Determine whether the given measures can be lengths of the sides of a triangle.
1. 8, 9, 17
2. 11, 12, 15
3. 10, 16, 8
4. 5.7, 9.2, 4.3
Determine Possible Side Lengths
Find the range for the measure of the third side of a triangle given the measures of two sides.
 Subtract the two given numbers
 Add the two given numbers
 Write as a range: (subtracted amount) < n < (added amount)
1. 7, 8
2. 12, 9
3. 6, 11
4. 45, 78
Review for Quiz (Targets 5a – 5b)
Perpendicular Bisectors
Find DL if DK = 26
and LK = 10
Angle Bisectors
Find UD if PC = 4
and UP = 7
Medians
Find BG if XB = x – 1
and XG = x
Altitudes
Find x if
mBEA = 7x + 6
Homework: 5c, page 251 from 5.2 in text (10 – 15),
AND page 264 from 5.4 in text (14 – 37) (30 problems)
Target 5d
Solve problems using inequalities involving one and two triangles.
SAS Inequality / Hinge Theorem – If ______ sides of a triangle are congruent to two __________
of another triangle and the __________________ angle in one triangle has a _________________
measure than the included angle in the other, then the third side of the first triangle is
__________________ than the third side of the second triangle.
Directions: Write an inequality relating the given pair of segment measures.
a)
and
b)
and
c)
and
Directions: Write an inequality to describe the possible values of x.
Hint: When writing your inequality, start with the largest x value on the left.
a)
b)
SSS Inequality Theorem – If two ___________ of a triangle are congruent to _________ sides of
another triangle and the _______________ side one triangle is _________________ than the third
side in the other then the ________________ between the pair of congruent sides in the first
triangle is ____________________ than the corresponding angel in the second triangle.
Directions: Write an inequality relating the given pair of angle measures.
a) WZV and UZV
b) PNS and MNS
c) GJH and XZY
Directions: Write an inequality to describe the possible values of x.
Hint: When writing your inequality, start with the largest x value on the left.
a) RPA = 4x + 16 and BPA = 7x – 5
b) ACD = 9x + 2 and ACB = 3x + 14
Homework: 5d, page 270 – 271 from 5.5, (3 – 4, 10 – 17) (10 problems)