Sherlock Inverted

Geometry Level 2

If the $\color{#D61F06} { \text{red angle} }$ (ACB) is a right angle, which of the following claims is not necessarily true?

Sherlock Holmes inverted: "Once you eliminate the provably true, whatever remains, no matter how probable, must be the falsehood."

The

\color{#3D99F6} { \text{blue angle} }

(BAD) is a right angle More than one of these claims is not necessarily true. The

\color{#EC7300} { \text{yellow angle} }

(ADE) is an obtuse angle The

\color{#BA33D6} { \text{purple angle} }

(ACD) is a right angle The

\color{#20A900} { \text{green angle} }

(ABC) is an acute angle

1 solution

Adam Phúc Nguyễn
Sep 25, 2015

Because $\angle BAD =90^{\circ}$ , so $\angle \color\blue{BAD}$ is also $90^{\circ}$ due to supplementary angles.
Because $ABC$ is a right triangle, then $\angle ABC < 90 ^{\circ}$ due to sum of interiors angles of a triangle.
Because $\angle ADC < 90^{ \circ}$ , $\angle ADE > 90^{\circ}$ due to supplementary angles.
AC is the perpendicular bisector of BD, so if $\angle B, \angle D$ gets larger, $\angle A$ will get smaller, and vise versa, but not $45 ^{\circ}$ .

Only the $\angle \color\blue{BAD}$ is not necessary true.