CHAPTER 10 / ESSENTIAL GEOMETRY SKILLS 369
Concept Review 2
- 360°
- 1,080°
- Draw a diagram. If the measure of ∠Ais 65°and
the measure of ∠Bis 60°, then the measure of ∠C
must be 55°, because the angles must have a sum
of 180°. Since ∠Ais the largest angle, the side op-
posite it, BC,
___
must be the longest side.- 60°. Since all the sides are equal, all the angles
are, too. - No, because an isosceles triangle must have two
equal angles, and the sum of all three must be
180 °. Since 35 + 35 + 60 ≠180, and 35 + 60 + 60 ≠
180, the triangle is impossible. - Your diagram should look something like this:
a°b°x° c°
a + b = c- If a triangle has sides of lengths 20 and 15, then
the third side must be less than 35 (their sum) but
greater than 5 (their difference). - No. The sum of the two shorter sides of a triangle
is always greater than the third side, but 5 +8 is
not greater than 14. So the triangle is impossible. - 25°and 130°or 77.5°and 77.5°
- Draw in the line segments PQ, PR, PS,andPT.
Notice that this forms two triangles, ∆PQSand
∆PRT.Since any two sides of a triangle must have
a sum greater than the third side, PQ+PS>QS,
andPR+PT>RT.Therefore,
PQ+PR+PS+PT>QS+RT.
Answer Key 2: Triangles
SAT Practice 2
- A IfAB=BD,then, by the Isosceles Trian-
gle theorem, ∠BADand∠BDAmust be
equal. To find their measure, subtract
50 °from 180°and divide by 2. This
gives 65°. Mark up the diagram with
this information. Since the angles in
the big triangle have a sum of 180°,
65 + 90 +x=180, so x=25. - 500 Drawing two diagonals shows
that the figure can be divided into
three triangles. (Remember that an
n-sided figure can be divided into n
−2 triangles.) Therefore, the sum
of all the angles is 3 × 180 °= 540 °.
Subtracting 40°leaves 500°. - B The third side of any triangle must have a
length that is between the sum and the differ-
ence of the other two sides. Since 16 − 9 = 7
and 16 + 9 =25, the third side must be between
(but not including) 7 and 25.
4. A Since the big triangle is a right
triangle,b+xmust equal 90. The
two small triangles are also right
triangles, so a+xis also 90. Therefore,
a=band statement I is true. In one
“solution” of this triangle, aandbare 65
andxis 25. (Put the values into the diagram
and check that everything “fits.”) This solu-
tion proves that statements II and III are not
necessarily true.
5. C IfAD=DB,then, by
the Isosceles Triangle
theorem,the angles
opposite those sides
must be equal. You
should mark the
other angle with an
xalso, as shown here. Similarly, if DB=DC,then
the angles opposite those sides must be equal also,
and they should both be marked y.Now consider
the big triangle. Since its angles must have a sum
of 180, 2x+ 2 y=180. Dividing both sides by 2 gives
x+y=90. (Notice that the fact that ∠ADBmea-
sures 100°doesn’t make any difference!)
50 °
x°A
B
C
D
65 ° 65 °25 °130 °a°
b° c°d°40 °
b°x° a°B
D
x°^100 ° y°
A Cx°y°