Question 937612
What you meant is
{{{CAT}}} is an isosceles triangle whose vertex angle is {{{"< A"}}}. The {{{m<C=40^o}}}. What is true about {{{DELTA}}}{{{CAT}}}?
A) it is an obtuse triangle 
B) {{{m<T=40^o}}}
C) AC = AT
D) all of these are true
Can you tell me why A or B? and what {{{m}}} means?
 
The triangle looks like this:
{{{drawing(300,200,-5,5,-1,4,
triangle(-4,0,4,0,0,3.3564),
red(arc(-4,0,3,3,-40,0)),
locate(-3.5,0.6,red(40^o)),
locate(-4.1,0,C),locate(3.9,0,T),
locate(-0.1,3.7,A)
)}}}
 
A symbol looking sort-of like {{{DELTA}}} is used to mean "triangle",
so {{{DELTA}}}{{{CAT}}} means {{{triangle}}}{{{CAT}}}.
A symbol looking sort-of like {{{"<"}}} is used to mean "angle",
so {{{"< A"}}} means angle A (angle CAT);
{{{"< C"}}} means angle C (angle ACT), and
{{{"< T"}}} means angle T (angle ATC).
The {{{m}}} in front of the {{{"<"}}} symbol means "measure of",
so {{{m<C=40^o}}} means the measure of angle C is {{{40^o}}} .
 
An isosceles triangle is one that has two "congruent" sides (two sides of equal length).
The other side is called the base.
The angle made by the two congruent sides is called the vertex angle.
The other two angles (the ones adjacent to the base) are called base angles.
The base angles in an isosceles triangle are congruent (they have the same measure).
 
So, {{{"< C"}}} and {{{"< T"}}} are the base angles, and both measure {{{40^o}}} , so "{{{highlight("B )")}}} {{{m<T=40^o}}} " is true.
 
Since the measures of the angles of any triangle add up to {{{180^o}}} , the measure of the vertex angle is
{{{m<A=180^o-("m<C" + "m<T")=180^o-(40^o + 40^o)=180^o-80^o=100^o}}} .
Since angle {{{A}}} , the vertex angle, measures {{{100^o>90^o}}} ,
it is an obtuse angle,
and a triangle wit one obtuse angle is called an obtuse triangle,
so "{{{highlight("A )")}}} it is an obtuse triangle" is true about {{{DELTA}}}{{{CAT}}} .
 
"{{{highlight("C )")}}} {{{AC = AT}}} is true because the definition of isosceles triangle is that the two sides that form the "vertex angle" are congruent (have the same measure).
In this case, the two sides that form the vertex angle
(the ones that end at vertex {{{A}}} ) are {{{AC}}} and {{{AT}}} .
 
So, "{{{highlight("D )")}}} all of these are true" is true about {{{DELTA}}}{{{CAT}}} .