Fill in the blanks to make the statements true.

(a) In right triangle the hypotenuse is the ______ side.

(b) The sum of three altitudes of a triangle is ______ than its perimeter.

(c) The sum of any two sides of a triangle is ______ than the third side.

(d) If two angles of a triangle are unequal, then the smaller angle has the ______ side opposite to it.

(e) Difference of any two sides of a triangle is ______ than the third side.

(f) If two sides of a triangle are unequal, then the larger side has ______ angle opposite to it.

(a) In the right triangle, the hypotenuse is the longest side.

Explanation: The longest side is the side opposite to the largest angle and in a right-angled triangle the largest angle is 90° and the side opposite to it is the hypotenuse.

(b) The sum of three altitudes of a triangle is less than its perimeter

Explanation: Consider ΔABC with altitudes as AP, BQ and CR on segments BC, AB and AC respectively

Consider ΔAPC

∠APC = 90° … AP is altitude

∠ACP is some acute angle less than 90°

Hence AC > AP … (i)

Similarly we can prove

BC > CR … (ii)

AB > BQ … (iii)

Adding (i), (ii) and (iii) we get

⇒ AC + BC + AB > AP + CR + BQ

⇒ sum of sides > sum of altitudes

(c) The sum of any two sides of a triangle is larger than the third side.

Explanation: if the sum of two sides is equal to the third then the points will be collinear and these points won’t form a triangle

(d) If two angles of a triangle are unequal, then the smaller angle has the smaller side opposite to it.

Explanation: The side opposite to larger angle is larger. And the side opposite to shorter angle is shorter. So, the smaller angle has smaller side opposite to it.

(e) The difference of any two sides of a triangle is less than the third side.

Explanation: if a, b, c are sides of a triangle we know that the sum of two sides is greater than the third

i.e. a + b > c

rearranging we get a > c – b

(f) If two sides of a triangle are unequal, then the larger side has larger angle opposite to it.

Explanation: consider ΔABC with AC the longer side than AB

Construct a line BD to AC from point B such that AB = AD

The angles are as marked

We have to prove that (x + z) > y

From figure

⇒ x + ∠BDC = 180° … linear pair of angles ∠ADB and ∠BDC

⇒ ∠BDC = 180° - x

⇒ z + y + ∠BDC = 180° … angles of ΔBDC

⇒ z + y +180° - x = 180°

⇒ z + y = x

⇒ x > y

⇒ ∠ABD >∠BDC

If x is greater than y then x plus something will obviously be greater than y

⇒ x + z > y

⇒ ∠ABC >∠ACB

Hence proved