Chapter 6, Problem 10CR

a

To determine

To find: The given statement is always, sometimes or never be true.

Answer to Problem 10CR

The given triangle will be sometimes isosceles

Explanation of Solution

Given information:The following statement is given:

“If a triangle is obtuse, it is isosceles”

The triangle is obtuse….. given

In obtuse triangle one angle would be ${90}^o<x<{180}^o$ , therefore other two angles may be equal.

Thus, obtuse angle triangle can be sometimes isosceles

Hence, given statement is sometimes true

b

To determine

To find: The given statement i.e. “The bisector of vertical angle in a scalene triangle will be perpendicular to base” is always, sometimes or never be true.

Answer to Problem 10CR

The statement will never be true.

Explanation of Solution

Given information

The following statement is given:

“The bisector of vertical angle in a scalene triangle will be perpendicular to base”

Since in scalene triangle all the sides are unequal

Thus, bisector of vertical angle in a scalene triangle cannot perpendicular to base

Hence, given statement is never true

c

To determine

To find: The given statement i.e. “If one of the diagonals of the quad is perpendicular bisector to the other, then the quadrilateral is kite” is always, sometimes or never be true.

Answer to Problem 10CR

Given statement is true

Explanation of Solution

Given information

The one of the diagonals of a quadrilateral is perpendicular to another diagonal

Since, one of the diagonals of a quadrilateral is perpendicular to another diagonal

Thus, quadrilateral will be kite

Hence, given statement is always true

d

To determine

To find: The given statement is always, sometimes or never be true.

Answer to Problem 10CR

Given statement is always true

Explanation of Solution

Given information Four points A, B, C and D are non - coplanar

One line segment is perpendicular to other line segment, i.e., $\overline{\text{AB}}\perp \overline{\text{BC}}$

One line segment is perpendicular to given third line segment, i.e., $\overline{\text{AB}}\perp \overline{\text{BD}}$

Since, points A, B, C and D are non-coplanar

  $\overline{\text{AB}}\perp \overline{\text{BC}}$ ……..given

  $\overline{\text{AB}}\perp \overline{\text{BD}}$ …….given

Then, Line segment AB will always be perpendicular to the plane formed by the points B, C and D because, lines coming out from the foot of the line segment perpendicular to plane, are always perpendicular

Hence given statement is always true

e

To determine

To find: The given statement i.e. “Two parallel lines determine a plane” is always, sometimes or never be true.

Answer to Problem 10CR

The given statement is always true

Explanation of Solution

Given information

Two parallel lines

Two parallel lines never meet even on extending

Thus, they form plane

Hence, given statement is true

f

To determine

To find: The given statement i.e. “Plane that contains two skew lines are parallel” is always, sometimes or never be true.

Answer to Problem 10CR

The given statement is never true

Explanation of Solution

Given information

The following statement is given:

“Plane that contains two skew lines are parallel”

Skew lines are the non-coplanar and do not intersect

Thus, skew lines can not be in two different parallel planes

Hence, given statement is never true

g

To determine

To find: The given statement i.e. “Supplement of complementary angles are congruent” is always, sometimes or never be true.

Answer to Problem 10CR

The supplement of complementary angles is always congruent

Explanation of Solution

Given information

The following statement is given:

Supplement of complementary angles are congruent

Two angles are complementary when their sum is $90°$ and complementary angles are always congruent then their supplement will also be congruent

Hence given statement is always true